Rod of Ages is an item you always want on Amumu. The Health makes you a bit more tanky and mana is an issue on Amumu that can be fixed with this item.

Sorcerer's Shoes Are all I ever get on Amumu. I wouldn't recommend Ionian Boots of Lucidity if you need CDR but rather some of the CDR situational items.

Stalker's Blade - Runic Echoes Makes your abilities hurt a lot more for ganks, and the passive can help in the jungle (and a few times has gotten be some lucky kills). If you have your mind set on one of the situational items then you can sell it though I keep it for most games.

Rylai's Crystal Scepter The main thing about this item that makes it nice on Amumu is the extra health which helps you survive when ganking. The passive is a nice touch for when chasing down enemies.

Liandry's Torment has been changed a bit from what it used to be. It used to be Magic Penetration option number 2, however now it doesn't provide Magic Penetration, but instead a decent bulk of AP, 300 HP, and two passives which are essentially made for fighting high HP tanks.

Void Staff is the other option you have of the 2 Magic Pen items for AP Amumu. If their team has a lot of MR, preferably 2 or more tanky-champions with a lot of MR, then you can buy this item instead of Liandry's Torment. This item has no +health so it's a risky choice if they have a fed burst champ, but the AP stat along with the passive will help you deal with any tanks on the enemy team, as well as improve your overall damage.

Banshee's Veil has received a rework making it a great item for AP Champions. Not only does it protect you from CC, but it also provides a generous amount of AP, CDR, and MR to counter any fed AP on their team. This item should be the first thing to look to if you are ever having trouble with an AP on the enemy team.

Abyssal Mask is a pretty solid AP-Tank item for AP Amumu as well as Tank Amumu. The stats really spike your survivability and the passive make it suitable for AP Amumu. This item is ideal if they have a fed AP champion, and Abyssal Mask along with Banshee's Veil can really make you hard to kill with all the MR adding up.

Rabadon's Deathcap is a pure damage choice and isn't required on Amumu for him to get kills. I only buy it if the game is one of those 40 minute ones and I have a lot of extra gold.

Morellonomicon is an item I used to always buy on Amumu. I don't really like it on him since it provides no health but if you miss your Bandage Toss a lot then the extra CDR can sometimes secure you the kill. If they have an annoying Soraka or Janna then you can buy it for the Grievous Wounds effect.

Hextech Rocketbelt is a pretty good item on Amumu. If you're chasing down an enemy and your Bandage Toss is just barely out of reach, then you can pop it to save your Flash and secure the kill. The health is great too, so if you're dying a lot of then get this item over Liandry's Torment.

Zhonya's Hourglass can save your Ashe if you get ulted on by a Katarina or somebody with a similar ult. You can also buy it for the extra Armor if they have a lot of Attack Damage

Pros:
1. Huge damage.
2. Game changing ult, if used right can get you some multi kills.
3. Fun champion (IMO).
4. Good at securing kills.

Cons:
1. Bandage Toss somewhat hard to land. (Can't go through minions)
2. Can be countered in jungle, and you can get camped there if your team wont help.
3. Must be fed to do much with an AP build.
4. Has a hard time clearing jungle.
5. Without Rod of Ages Amumu runs out of mana easily.

This is the path you must take when jungling. Always start at Blue Sentinel. Amumu relies heavily on mana in the jungle. Next go to Greater Murk Wolf. If you need time to regenerate health, then kill the Rift Scuttler before finally moving onto the Crimson Raptors. While jungling, kill the Rift Scuttler if you are at 20-50% health or mana before moving onto other camps, otherwise just Recall. Try and only use Smite if you absolutely need to. (It really helps)

Update: Since all the monster camps now spawn together, you can clear a bit more dynamically as Amumu. For example, if you have a selfish team that isn't all that fond of leashing, then you may want to grab Tantrum instead of Despair so that you can clear the Crimson Raptors. Just don't be too quick to grab Despair, though even if you do, Despair should probably work for Crimson Raptors too.

I would not recommend using this build for Ranked. It's a fun build for Unranked but for Ranked you should go Tank Amumu.

If an enemy turret dives then Bandage Toss them, or use Curse of the Sad Mummy if you are really low. You want to position yourself so all the enemies in teamfights get hit by Curse of the Sad Mummy and this can really help out your team.

Bandage Toss costs tons of mana so I'd only recommend using it if you have Crest of Insight or if you absolutely need it to kill a monster. Amumu is the god of Crimson Raptors and can kill them by toggling on Despair and then mashing your Tantrum over and over so remember to stop by their nest whenever they're up. You can kill the Gromp after your first Recall but as for starting off, you want to avoid him and go Blue Sentinel > Greater Murk Wolf and then Crimson Raptor before recalling.

Amumu did not receive any major changes, in fact you can pretty much still play him the same way as you did before. However the rework did in fact buff him. His Cursed Touch passive really leans toward favoring his AP build since it can make your abilities hurt that much more, and even help your AP mid laner or other AP teammates shred down the enemy during a gank.

The only technique I can really highlight in playing Amumu with his new rework is putting in a bit more effort in landing his Cursed Touch on enemies. An example is instead of Bandage Tossing to engage, you can try and auto attack to engage, and then blow your abilities for that 15% true damage. Don't fixate on his Cursed Touch too much though. If the enemy is getting away, just Bandage Toss anyway rather than giving up once you can't apply Cursed Touch. Yes, Cursed Touch helps but it isn't necessarily needed for Amumu to do damage.

Little baby Amumu is usually played as a jungler, though as a jungler, I personally tend to focus more on kills rather than CS, as it is hard to get tons of CS when stuck doing jungle camps. CS really helps you get an S so to ensure you get CS as well as kills, it is best to go top lane Amumu for getting your mastery up. You can use the same AP build as a top laner, so there is no need to go a traditional tank Amumu. If you are bad at laning and would rather get your mastery up through jungling, you can always take out some of your time to go into a lane and farm for a bit, though sometimes teammates might get pissed if you take their CS early game so I'd recommend doing this later on past laning phase.

As you can see, a match where I went jungle Amumu I did quite well on kills, though lacked much CS which lead me to only getting an A+ by the end of the game.
In a game where I went top Amumu, I finished with a similar K/D/A though because I had ended with just more than double the CS of the jungle game, it drastically affected my grade and rather than getting a mere A+ I got an S+.

Getting to level 6 with Amumu was a real challenge for me back when I didn't know about how much CS can affect your grade, and I spent the entire week spamming several games with Amumu hoping to pull off an S- at the very least. Though understanding what can affect your grade helped me get to level 7 in a mere 4 games, and so I hope this helps you get there too 😉.