Amumu is a great champ to jungle with. With a wide range of items that compliment his kit, you can build him to be extremely tank or an off-tank with a lot of damage.

He is decent at early ganking, but he can truly shine mid to late game at team fighting. His ultimate, Curse of the Sad Mummy and Bandage Toss give him some powerful Crowd Control. His ultimate synergizes extremely well with other AoE spells such as Command: Shockwave, Bullet Time or Solar Flare.

Amumu does have one of the slower timed skill shots in the game with Bandage Toss, so learning to use it well can take some practice.

He does have mana issues, so learning efficient use of Despair is important.

Overall, I find Amumu an extremely fun champ to play, who brings a lot of utility to your team.

Like every champion, Amumu has strengths and weaknesses. The better you understand them, the easier it is to maximize his kit to hurt your enemies while planning ahead to mitigate your weaknesses.

PROS:
+ High AoE damage
+ Powerful CC & high damage Ult
+ Relatively quick clear time
+ Strong CC initiation
+ Tanky with high damage

CONS:
- Blue buff dependent in jungle
- Bandage Toss takes some getting used to due to its slow speed
- Susceptible to counter-jungling
- Ultimate Curse of the Sad Mummy has a high cold down

With the new Enchanted Armor the armor runes give 1 extra armor immediately and are extremely helpful with early game survival.

Greater Mark of Armor is my choice of mark runes due to the S4 changes with the higher damage from jungle creeps. With Cursed Touch, you don't have to use Greater Mark of Magic Penetration for early game damage.

Greater Quintessence of Health for increased early survivability. Again, with the S4 changes, early survivability is extremely important. As such, I think these runes offer the most benefit to Amumu

Scaling MR runes as AP damage is usually lower early game. Flat MR runes are a decent alternative.

Defense:
Block & Unyielding are excellent choices for damage reduction.
Enchanted Armor works well for this build as you will be building a lot of armor and MR.
Juggernaut is a powerful choice as most of the items built for Amumu give health.
Hardiness & Resistance for increased survivability.
Reinforced Armor for late game survival. This can add up to a lot less damage taken.
Tenacious is a must have. In a team fight, this is an extra 20 armor and 10 MR.
Legendary Guardian combined with your tenacity item, this gives you a 50% reduction to incoming CC.

Utility:
Fleet of Foot is equal to one move speed quint. Best 3 points you can spend in the first tier.
Summoner's Insight Summoner spells are extremely powerful and being able to use them more often is amazing. With the changes to Smite, this gives it a very low CD to help with faster clear times. This is important when factoring in the new jungle camp.
Alchemist gives you an extra tick on your health pots.
Runic Affinity is a must have as Amumu is very blue buff dependent early on.

Cursed Touch - Your auto attacks give you free magic penetration. Always keep this debuff up on your target.

Bandage Toss - Your primary move to initiate a fight or to prevent someone from running. Early game, it's often a good idea to wait to use this to prevent your enemy from running away. For team fights, use this to position yourself for using your ult.

Despair - Incredibility powerful ability, as it does damage based on the enemy's maximum health. This ability uses mana per second while active, so make sure to pay attention to toggling it on/off, otherwise, you will run out of mana.

Tantrum - Passive gives a nice increase in survivability and the active does a moderate amount of AoE damage.

Curse of the Sad Mummy - Your Ultimate, which is truly awesome. It's range is as big as the auto attack range of most ranged champions(500 radius, 1000 diameter), it has a high AP ratio, so great damage, and it roots enemies for 2 seconds. Downsides are the relatively high mana cost and long cool down.

With the new jungle changes, clear times are slower, so leveling Despair first helps compensate for this.

You want to level Bandage Toss to lower the cool down. This will help a lot with securing kills.

With the armor runes and defensive masteries, the damage reduction from Tantrum isn't as important, so we level it last.

Curse of the Sad Mummy is your top priority to level at 6, 11 & 16.

The TAB key is your friend. Pay attention to what everyone is building. If someone else on your team is building a Locket of the Iron Solari, you don't need to build one. You also need to keep track of what the enemy is building and be willing to adjust accordingly.

Stealth Ward - no more excuses. Plant wards. Just do it. Late game, you will probably want to trade this for the Oracle's Lens.

Hunter's Machete - standard starting item for jungling.

Mobility Boots - I take these 90% of the time. These will let you have greater ganking potential and map control. If you need more survivability early on, get Ninja Tabi. If you're really ahead and want more damage, get Sorcerer's Shoes.

Spirit of the Ancient Golem - must have item. Bonus health, regen, Tenacity and CDR. This is one of the best items for the price you can buy for Amumu. He greatly benefits from everything about this item.

Sunfire Aegis - provides armor, health and AoE damage. Essential item!

Abyssal Mask - in addition to the AP and MR, the aura on this will benefit you and everyone on your team that does magic damage. Just by standing in the middle of the enemy team, you will cause your team to deal more damage.

Rylai's Crystal Scepter - one of my favorite offensive items on Amumu. With this item, Despair constantly applies the slow effect to everyone around you.

Liandry's Torment - Also pairs very well with Despair. Combined with Rylai's Crystal Scepter, you will cause a ridiculous amount of damage.

Warmog's Armor - an amazing item on any tank. If you need to be extremely tanky, take this over Liandry's Torment.

Randuin's Omen - if you have an out of control ADC, this is a great solution to help shut them down.

Thornmail - if the enemy team is almost entirely physical damage based, this will shut them down.

Spirit Visage - if you're facing a very AP heavy team, this is a great choice. The CDR is also a tremendously useful stat.

Locket of the Iron Solari - the stats on this item are awesome for Amumu, but because this item is so frequently build by support players, it's not part of my core build. If no one on your team is building this, pick it up as one of your last two items.

With the addition of the new jungle camp and the changes to Smite, the routes and priorities have changed.

As Amumu, you pretty much have to start with blue buff, or you'll run out of mana too fast.

Make sure to ask your team(politely) to watch the red buff. With the new vision wards, use yours to make sure no one is coming to Smite steal over the wall and ask someone to ward the red buff to watch for counter-jungling.

With the increased stats on the jungle monster, use Smite on blue to reduce the amount of damage you take and increase your clear time. With CDR from masteries, you have a 36 second CD on Smite. You should be using it a lot. Just watch the clock to make sure you have it available to help secure buffs and Dragon.

From Blue, go around the jungle in a circle, adding in the new camp.

In order: single large wraith, wolves, small wraiths, lizard elder. After getting the red buff and level 4, you need to look to gank. If people only used the free ward from the trinket, it should have expired by now.

GANKING - as a jungler, this is where you change the game. Make sure to use pings to let your team know where you are going so they can help set you up. If a lane has been pushed into the enemy tower, don't try to force a gank. If you think it likely that the enemy jungler might go after the pushed lane, go sit in the bush by that lane for a few seconds and see if you can ambush the ambusher.

When you gank and have red buff, you can often run in behind the enemy and save your Bandage Toss to stop them from running away.

I disagree with mid-laners that think blue buff always belongs to them. The Ancient Golem is also with a lot of XP and gold, which you need. With Amumu, I usually take the first 2 blue buffs to ensure a solid early jungle and then give the rest of them to the AP carry.

Early game, don't be afraid to use your ult to secure a few kills for your team. Yeah, it's a long CD, but during the laning phase, you need to help your team get all the kills it can. You can also use Flash to pop in and Ult to set your team up for kills.

You also need to pay attention to when your laners need to go back to base. If you can, cover their lane and pick up a few minion kills to prevent damage from going onto their tower.

DO NOT TOWER DIVE EARLY GAME. Not sure I can say this enough, but a low levels, you are not super tanky yet and the tower will destroy you. This will probably result in giving the enemy a double buff, giving them a massive advantage in their lane. Just don't tower dive early on. It's not worth the risk. If you loose your buffs, your ability to jungle takes a huge hit.

Early game, you need to help your team force the enemy to burn Flash and set up kills to help win lanes. With the changes to tower gold, you want to be there when the outer towers die, so pay attention to tower health.

Make sure to use pings to coordinate with your team when you are planning to initiate a fight. When you dive in and use Curse of the Sad Mummy, you want your team ready to kill the champs you just trapped.

As a tank, don't be afraid to take damage. This is not a license to eat skill shots to the face, but if you can force the enemy to focus their damage onto you and not your carries, you're doing your job.

I consider it a huge win to be the only person on my team to die in exchange for 3 or more of the enemy team dying.

While it's your job to initiate fights, also make sure to watch out for your carries. They are very squishy and need help to stay alive. Help peel that annoying Lee Sin off of them so they can kill the enemy team.

Amumu is a great champion that bring a great kit to your team.

Make sure to communicate with your team to set up early ganks and team fights as you transition to mid and late game.

Wards are always a great choice. Use them and your trinket frequently.

Make sure to frequently glance at the mini map to see where people are & what enemies are missing. This will help you make better choices about where you need to be.

I hope this helps you with learning how to play Amumu.

Cheers :)