Amumu had been played a lot more often but he is still a very underplayed champion. He's a tank/disable champion that is very useful in team battles.
Bandage Toss - Amumu tosses a sticky bandage at a target, it stuns and damages the target while he pulls himself to them. Throws a bandage to target location. If it hits an enemy unit, Amumu will pull himself to the enemy, stun them, and deal 80 / 140 / 200 / 260 / 320 (+1 per ability power) magic damage.
Cooldown10 secondsCost130 ManaRange 1,100
Despair - Overcome by anguish, nearby enemies lose a percentage of their maximum health each second. Toggle: Nearby enemies are drained by 2 / 3 / 3 / 3 / 4 (+0 per ability power)% of their maximum health each second.
Cooldown1 secondsCost10 Mana per SecondRange 300
Tantrum - Permamently reduces the physical damage Amumu would take. Amumu can unleash his rage, dealing damage to surrounding enemies. Each time Amumu is hit, the cooldown on tantrum is reduced by 1 second. Passive: Amumu takes 1 / 2 / 3 / 4 / 5 reduced damage from physical attacks.
Active: Amumu deals 155 (+1 per ability power) magic damage to surrounding units. Each time Amumu is hit, the cooldown on Tantrum is reduced by 1 second.
Cooldown20 secondsCost50 ManaRange 400
Curse of the Sad Mummy - Amumu entangles surrounding enemy units in bandages, causing them to be unable to move or attack and take damage each second. Amumu entangles surrounding enemy units, causing them to be unable to move or attack for 3 seconds, dealing 50 / 100 / 150 (+0 per ability power) magic damage to them each second.
Cooldown180 secondsCost100 / 150 / 200 ManaRange 600 Masteries
In my opinion, I went straight for utility and tank. The cooldown reduction in the utilities is great and the damage reduction against creeps is good for early game.
For this build specifically, go for all cooldown reduction. Amumu is an anti-tank for the champions with massive HP.
Go for Heal and Teleport. It may seem noobish but I'll explain later. You can also replace Teleport with Flash or Cleanse. Rally is also very good.
1- Doran's Shield and Mana Potion
2- Chain Armor -> Giant's Belt -> Sunfire Cape
3- Boots of Swiftness
4- Sunfire Cape
5- Abyssal Scepter
6- Sunfire Cape
7- Sell Doran's Shield and get Guardian Angel
17- 18. Bandage
How To Play and Reasoning
Early game: Level 1-5
Start off by last hitting creeps with default attack and tantrum. If it is safe, tank the creeps so you can tantrum faster. Use Mana Potion if needed. Once you hit Level 5, you should have at least 700 gold. Use that and buy a chain mail. Now you can solo Golem using despair and tantrum. Once you killed Golem you can leave despair on or just tantrum spam the creeps. Kill Golems whenever they respawn. Even in the enemy's Golem. The cooldown reduction is deadly with despair. With runes and Golem Buff, you should have nearly 40% cooldown reduction which is the maximum. Once you get your first sunfire cape you can solo dragon.
Mid Game: Levels 6- 13
Now to explain why I have teleport and heal. For early team battles that require you're help but you are not there, use teleport. Also you are moving around constantly for the golem and dragon. It is very often that team battles start without you since you are always busy pushing. The heal is for healing of course. Since you will be initiating, everyone will focus fire you. Use that heal and then use Curse of the Sand Mummy. I don't like to initiate with Bandage because I'm not very accurate with it and there are always creeps in the way. So after you initiate with your ult, make sure you have despair on and always tantrum. If the opponent is still living, use bandage if they are running away.
Late Game: Levels 14-18
Now you should barely die. The only weakness you have are against magic users but you have some health to back it up. Your sunfire capes should be raping the squishies and you should now be nearly invincible. If you are to die, you have your guardian angel. You can get your guardian angel sooner than your sunfire capes but I prefer to get sunfire capes first to push.
Bandage can be used both for initiating but I prefer to use it for people that are running away. Also Rally is a good substitute for heal since it lasts longer and you'll most likely be fighting around the area it is casted.