This is a build for pro Amumu trying out a new build. I don't recommend new comers trying it.

I played about 400 games with Amumu this works best for me. I do play 3v3 though so take that mind. I try to get boots of swiftness as soon as the enemy team has there boots completed. If they don't have it yet I keep building the Aegis of Legion. If they do I rush Boots of Swiftness. After Aegis of Legion remember to buy one Sapphire Crystal, this is very important. The get Giants belt first and finish up Sunfire Cape. Glacial Shroud will give you tankiness and the ability cooldown and mana Amumu needs. You will be able to have Curse of the Sad Mummy in every team fight, wich means you win. Get a Negatron Cloak and after finish up the Frozen Heart. If the match last that long get Abyssal Scepter and finish up with Rabadon's Deathcap. At this point you could single handely take out all squishys.

I do pay little attention to Despair until the last levels. When your'e at level 4 is when I recommend you start being more aggressive. You will have 2 points on bandage and tantrum. If you see a clear opening and your team mate is around to help out that's when you strike with bandage.

I go with Flash and Ghost in everygame. Flash to ult in on team figts or to escape. Ghost helps in those situations when people might escape all you need to do is stand next to them with you despair. Even better if you have Sunfire Cape.

My masteries are set up so I can stay in lane longer and have cooldown reduction. Since Amumu's Curse of the Sad Mummy has a long cooldown this helps.

The runes is pretty simple I do this to get a bit more tanky from the beginning. Since I'm getting Aegis of Legion first these runes help me maximize it more.

Overall this build helps me win 90% of the games. It gets me tanky/able to do damage while also supporting my team mates. If you need any proof I'll be glad to to show you my match history.