Okay I'm still not level 30 yet, I'm just a casual player. However, I discovered the fun of playing Amumu a couple months back when he was up for free. A couple of my friends have tried playing him and are having issues trying to be "as successful on Amumu" so I decided I'd toss up a build on here for them to work off of.

Keep in mind Amumu is weak early game despite being a tank. Most people I see play Amumu as a jungler. That's all fine and good, I've done it a couple times myself, but often you will run into a Warwick or Nocturne that really want that jungle to themselves. Amumu is playable in lane you just gotta be clever about it and work with your team.

This is my first build on here and I'm still learning how to do this stuff so please be patient. I'll brush this build up more as I go along and make it more streamlined and easier to use. Keep in mind your job as the tank is not to get the kills. It's to make yourself so annoying and enough of a threat that people -WANT- to hit you. That way your team can kill and survive.

If you have any suggestions/improvements for this build let me know.

For runes I usually feel like the extra mana regen on Amumu lets you stay out in the lane longer along with Clarity. Being in that lane as much as possible is vitally important as Amumu isn't exactly the strongest lane champ out there. The Insight runes for the MPen are pretty self explanatory. You want that damage getting through. The Quints are to give you that extra HP which is great at the start.

My masteries seem a little odd I know. I spread my points out a little bit to try and make what I see as the best use of Amumu's skills. Running out of mana is one of your worst enemies on Amumu. The mana regen out of the utility tree lets you keep despair up and fight longer. Improved exhaust is to slow those runners and get that extra damage in and also good to seriously cut the damage down on a melee carry. I even went down to the MPen mastery. It helps push more of that damage through. All champs have some natural magic resist. With this build and Amumu's passive you can shred through most people's magic res like it's nothing.

I know it seems odd to build that Philosopher's Stone. It has helped me out so much at the start of the game that it's ridiculous. Sunfire you will want as early as possible as it speeds your farming up immensely and works very very well in conjunction despair. It also gives some nice HP.

Warmog's is good on nearly any tank in my opinion. If want to stay in that fight as much as possible.
Banshee's Veil, pretty decent HP and Mana on the item plus magic resist. If is good to negate some initiation abilities letting you start the fight on your terms.

Randuin's Omen, I have just recently started to play around with. I used to build Thornmail mostly just because it was comical watching Xin Zhao slowly kill himself trying to fight you while Despair and Sunfire chewed away at him. I have found that the slow from Randuin's is great though.

Your boots are situational. If they have a lot of CC then I suggest using Merc Treads instead of Sorc Shoes. If you need the extra armor then Ninja Tabi works fairly well.

Tantrum is a great ability. It's a small burst AOE which can help in getting last hits in on Minions. The cooldown on it lowers as you get hit, and on top of that it lowers the amount of damage you take as a passive.

Bandage toss is your Initiate. Early game as soon as I hit level 2 I will often use bandage toss + Tantrum to harass. Bandage toss causes a minor stun which you can use to your advantage.
It is a skill shot though so it will take some practice. When you hit with it, immediately hit Tantrum and then click to run away. I have fought against this myself, you have no idea how annoying it is to barely get to do anything in retaliation while the little emo midget is running away.

Once you get to level 6 and Curse of the Sad Mummy, things start snowballing. Curse is a very powerful move. It slows, does some damage and think it actually causes a very momentary stun. I'm not sure on that though as I've never been caught in it myself. When the team fights start, your Ultimate can seriously change the tide of battle.

At the start of a team fight you will want to turn despair on. Then pick your target (preferably a squishy) then bandage toss in. As soon as you are in, I usually use Tantrum then Ult. That will have everyone stuck in the effect of Sunfire and Despair for a moment or two and you won't believe the amount of panic it can cause. If you have to your Ult works great for a last ditch escape as well as you can slow the whole enemy team.

Clarity is great for mana through most of the game as some of your abilities get rather costly in any type of prolonged fight. This will also help any mana dependent teammates you have early game as you will also give them close a small amount of mana back.

Exhaust is a very useful summoner spell. It slows the target and with the mastery to improve it, it lowers their resists. Their damage is also cut down. This helps when chasing or escaping.

I can't stress enough that when you are playing the tank you are there for your team not for the kills. Your job is to try to keep them alive as they are the main damage dealers. You give your support to them and set them up to get the kills. You'll get some in there with sunfire and despair occasionally slipping that last bit of damage in. Really though it's about the team. With this build you can also tank towers for your tower pushers pretty well. There are a few items that can be switched out based upon situation. If they are almost all AP I will usually switch out my Randuins for a Force of Nature