So don't hate, comment for any suggestions but I guarantee this will work if you are good at farming, and landing bandage toss. I took some info from Amumu: The Tank Slayer DannyMagick

Pros
Great damage
Good Defense
Good at escaping
Great at ganks
Great AoE damage
Great in teamfights
Supports other AP casters
If taking alot of damgage can use Zhobya's Hourglass
Guardian Angel saves you most of the time
Ok health
Amazing initiator
Amazing farming
Your spells heal you bandage toss at fullbuild will heal 200 and tantrum about the same
Cons
If CC'ed then you are useless and will probably get bursted down
Weaker aginst Ap magic damage champions
Can get burst down quickly if ganked
Squishy early game definitely against AP casters
Kill steals a lot
Gets blamed in team fights or in ganks if they fail for missing

It really doesn't matter what sequence you put it in just start out with all 3 spells at level 3 and grab your ult at lvl six, eleven and sixteen

Bandage Toss: Skill-shot, aim, click. You want to lead this a little bit and if you aren't close to a wall it won’t pull you through even if you hit your target. I often use this after my teammate that I lane with initiates so he can unload a full combo. You can also drop unexpected squishes with those unexpected impossibly far skill-shots followed by a Tantrum. Don't be upset if you miss your Bandage Toss when you’re first starting out as Amumu], as it is skill-shot that travels differently than some other skillshots. You’ll notice a difficulty in hitting with it if you’re new to Amumu and if you play champions like Dr. Mundo or Ezreal, whose skill shots move very quickly.

Despair: Amumu cries on his enemies doing percent damage per second + magic damage. Stacks very well with your Sunfire Cape and becomes even stronger once you get your Abyssal Scepter because of their reduced Magic Resist.

Tantrum: Activate to break dance and deal damage. Getting hit reduces the cool down. Passively reduces damage taken. Great farming skill, huge source of AoE damage. Stacks very well From Juffalo

Curse of the Sad Mummy: This is an amazing skill. When activated it will do a massive AoE snare for 2 seconds, the enemies take damage instantly and are completely unable to move for the ultimate's duration. Don't be afraid to use this skill, after a while you will know if it's a good time to activate it to keep them from running, or to let the lucky Master Yi run back to base with 14 health. This skill is nice to use in combination with Flash as you can aim where to drop your ultimate much better or push yourself a tiny bit farther into position to hit all 5 champions, rather than only 3.


Level your skills equally

Early-game:

Often I find myself in the scenario where someone has been harassing for a while early game and I decide to NOPE them harder than that one time they really wanted the cool action figure with Kung-Fu grip. I would Bandage Toss, cry my life off, throw a Tantrum, curse them, and if I timed it correctly I should be able to stun them again before they are able to run away leaving them with another Tantrum and a few more seconds of Despair. Depending on my laning partner they will be either super dead, really dead, or at 25 health. If they are extremely low health and you are still almost full you can go in behind their turret, Bandage Toss (don't miss) and either 1-hit them, or Tantrum and flash out of turret range.
Although this seems extremely unlikely it's really easy to pull off against morons that overextend, most of the time it's a good thing to let them push you to turret, just don't let them do significant damage to it. *Level 6 minimum*

Also, when you are facing heavy melee auto-attackers that are overextending, I.e. Master Yi, Tryndamere, Jax, you are able to Bandage Toss and then Tantrum and run back to safety without being damaged. This is a good way to harass with Amumu at a low level without taking much damage Use Tantrum to farm.

The entire goal for the early game as Amumu is to farm as much gold as possible and try to help your laning partner get as many kills as possible.

Mid-game:

During the Mid-game you want to focus on getting the Golem Buff as much as possible, ganking as many people as possible, and pushing as many turrets as possible. Team fight are a must in the Mid-game and you are the key to winning those. If you can time your Curse of the Sad Mummy properly your team should be able to take out at least 3 enemies in a team-fight. If things go bad you are pretty good at escaping due to your stun, ability to pull through walls, and do heavy damage to anyone dumb enough to follow you extremely closely.

End-game:

End-game, as Amumu you are pretty much the main target now, but if you are far enough into your build you should be strong enough to take a massive chunk out of turrets and enemies. Keep on pushing and keep on being a key part of the team fights is the key to victory. Amumu should be able to 1v1 any champion on the enemy team and by end-game can duo Baron Nashor with any other champion in the game. You are Amumu The Tanky AP Machine, wear the name proud and don't let anyone on the enemy team push you around.