Not Updated For Current Season
The masteries shown here are not yet updated for the current season, the guide author needs to set up the new masteries. As such, they will be different than the masteries you see in-game.
Why this ?
For me, Amumu is one of the best (or the best) tank in LOL. I played 50 games with him, and I always like him. To start, you must know that Amumu is not the character who goes to kill everybody in the start. In fact, in the first 10 minutes, generally, I have 0/3. It's not a good score, I know, but if you continue to buy what you need, you never go to die again, you could even kill the opponent at the end.
For the information, I'm french, so my english lvl is not the best.
How to play
Bandage Toss is really useful for the first blood, or to stop an opponent who tries to go away. If you play Blitzkrank, it's the same thing. But, it's a good idea to lvl up this spell, lvl 1 is very good.
After, you take the best spell of Amumu, Despair, what deals damage in function of the life of your opponent. More he has, more he loose.
Tantrum is useful for is passive effect, you can up your harmor.
To finish, Cruse Of The Sad Mummy is one of my favorite ulti in the game. With it, you can stun people next the tower, or stop every opponent and let the others doing damage. It's really useful, I promised.
Why those items
With Philosopher Stone you have health and mana regen, who is useful to don't back in the start, and to do Despair without loose to much mana.
The goal of the other items is to have armor and life, and moreover can do damage. In fact, With the Sunfire cape or Thornmail, you do damage to the opponent, and good damage. With your Despair, it's really strong.
For information, this build it's again physical damage team, for magic damage team, you can change the Torhnmail into Banshee's veil and one of the Sunfirecape into Force of Nature.
For the other informations, I leave find you the solution.