Amumu Amumu is an AP tank who can be played jungle. Despite being able to build full AP or full tank, I prefer a build that tries to mix both of them, in that way you can still be a huge frontline for your team while also having the DPS to try to take down carries. He's a reasonable duelist and the AOE damage in his kit allows him to have a fast clear and move around jungle pretty fast with his Q. However amumu really shines in teamfights with his ultimate ability who has the potential to change even the hardest of games.

Code:

Pros → fast clear → good ganks post 6 → can engage fights → Has a lot of crowd control → you can afford to fall behind a bit in xp/gold

Cons → weak early game → vulnerable to invades → Doesn't have an escape tool → it's not hard to counterpick → Like every tank, you need decent teammates around you

Cursed Touch
Amumu's passive makes his autoatacks and abilities to apply a curse that amplifies any magic damage that the oponnent recieves as extra true damage. This is the reason why amumu has great synergy with AP champions since they'll benefit the most from it. Bandage Toss
His Q is a dash that stops the enemy movement and gap closes amumu to the target and stuns them, it can very useful for ganks, to get yourself close to start a fight with your ultimate, and specially to move around the jungle faster, instead of walking up to each camp, you can easily just Q through walls to speed up your clear.
This is the skill you max 2nd. Despair
Amumu's W is an AoE ability who deals % max health damage, which makes it very helpful to clear the jungle. It has no cooldown since it's an activate/deactivate ability, so you always want to keep it up when you're clearing camps, just be careful not to run of mana! Tantrum
Amumu's E is divided into his passive and active. His passive negates damage coming from basic atacks from a flat amount and he active is an AoE ability who deals damage to nearby enemies. Whenever amumu gets hit by an autotack, the cooldown of this ability is reduced by 0.5 seconds, whcih is very helpful to clear the AoE camps in the jungle. Curse of the sad Mummy
Now this is the ability that makes amumu shine. It's one of the most impactful ultis in the game and can easily win a teamfight for your team. It's an AoE stun ability who stuns your enemies nearby. This is the ability that can make amumu shine and carry games. However your oponnents will be expecting it, so make sure you try to abuse fog of war to try to get a flank position with your Q to get an ulti to the enemy backline.

Conqueror:Despite being a tank, amumu is actually not a bad duelist, so with conqueror you can sustain through extended fights and make a good use of your AP scaling. Triumph:Due to his tank nature, he gets a lot of HP natures naturally, which means you get a lot of value in teamfights with the HP regen from triumph, the extra gold helps aswell. Legend: Alacrity: Like every tank however, they tend to have a slow first clear, so alacrity will give you some extra atack speed to make sure you can keep up with other champions. Last Stand: Coup de grace is a rune that doesn't make a lot of sense here since you're not an assassin, the same applies to cut down because you're more likely to have more Hp than the oponnent. Which means the only logical choice is last stand. Conditioning: Since you're going to stack a lot of resistances, getting that extra armor/magic resistance from runes is going to help you scale up a bit better and be a better frontline for your team. Unflinching: Despite being a tank, that doesn't mean it's fine to be cc locked for a few seconds, this is one of the best runes of the game, and it will give you enough tenacity to make sure you can play out fights.

Sunfire Aegis In my opinion this is the best tank item for amumu. Despite being more expensive, it provides more damage from it's passive and the tenacity from the mythic passive is really helpful aswell, you already have enough cc on your kit, so getting a little extra damage feels good when you're in fights.

Demonic Embrace Despite being a tank, this item synergizes perfectly with amumu, it gives % max health damage to combine with his W, it gives HP and a few resistances aswell, and amumu also scales really well with AP, so it's perfect for him.

Abyssal Mask The other magic resistance items aren't that valuable on amumu, you don't heal so spirit visage is useless, stacking MR can be good but it's not always optimal. Abyssal Mask however amplifies your damage and your allies' damage when you immobilize someone, which is perfect for amumu, since it can combine with his passive and amplify your teammate's damage by a lot and your own.

Force of Nature Despite not being the ideal item, if the enemy team is full Ap, this is a great situational item.

Thornmail I think this is a good core item, since he's a reasonable duelist and there's a lot of champions with sustain in the game, this will always make sure you can deny their stats and turn it around.

Randuin's Omen This is the perfect item vs crit champions and to survive the ADCs in the late game, it gives you nice stats and a very useful active to slow them down and reduce their damage to you.

Gargoyle Stoneplate There's no point in having HP if you don't have the armor/magic resistance to follow it up and vice-versa. This is the item to keep you alive when you jump into 5 people and activate it. you'll be ready to buy enough time to apply your CC and to be the frontline to your team.

Amumu can afford to start wherever he wants, even solo.
The only important thing to do is always full clear. You can choose to start solo at the raptors if you skill Tantrum first or if you want your team to leash you at the red or blue buff you start with Despair. Either way you need to make sure you clear all your camps before starting to gank. It's very important to get your level 6 powerspike as soon as possible since it's your best tool to gank and initiate fights.
Amumu has a weak early game so you want to avoid the enemy jungler usually.
He's also very decent to get drakes early, so after your first back if you're full HP and have smite up you wanna look to try to get dragons if you can, fighting around objectives or in the jungle is very important for amumu since the oponnents have less space to run away from your ultimate and it's easier to play with the fog of war.

I'll add more later