Cursed Touch (Passive)

This unit has reduced Magic Resistance.                               

Skill Order

1:  Tantrum
2:  Bandage Toss
3:  Tantrum
4:  Despair
5:  Tantrum
6:  Curse of the Sad Mummy (Ultimate)
7:  Tantrum
8:  Despair
9:  Tantrum (Max)
10: Despair
11: Curse of the Sad Mummy (Ultimate)
12: Despair
13: Despair (Max)
14: Bandage Toss
15: Bandage Toss
16: Curse of the Sad Mummy (Ultimate Max)
17: Bandage Toss
18: Bandage Toss (Max)

Reason for the skill order

TANTRUM IS KEY!!! Tantrum is huge harass and huge money. Rank 1 Bandage toss is all you need because the distance and time of stun is the same as max rank. You essentially get more out of Despair being ranked pre-bandage toss. Curse of the Sad Mummy at all available opportunities for obvious reasons.

Master Suggestion

I personally suggest 0/21/9.

Defense Tree.

3-Resistance
3-Hardiness
4-Evasion
2-Defensive Mastery
1-Nimblebess
3-Harden Skin
4-Veteran's Scars
1-Tenacity

Utility Tree.

3-Perseverance
1-Good Hands
4-Awareness
1-Greed

Again this is my personal suggestion but really Anything will work.

Rune Suggestion

I like to go mass hp runes again your runes are very personal it can even be ability power and work successfully.

Summoner Abilities

I have tested various amounts of summoner abilities and I finally broke it down to 2.
After testing with just about every ability I found that Ghost and Flash seem to be the best two but this all depends on playstyle again so its really up to you.

2: Items

Item Choice

Regrowth Pendant (starter)
into
Philosopher's Stone (This is good for A: gold and B: survival) (Able to sell late game)
Ninja Tabi (Move speed with Dodge. If your gonna tank do it right :P)
Sunfire Cape x 2 (I personally think getting 2x Sunfire Capes is huge. It gives you 80 damage from item and its got mad health and armor)
Warmog's Armor
Frozen Mallet

Other suggested Items are:

Warden's Mail (Good depending on the champioins you are fighting)
Atma's Impaler (After the 2x Sunfire Capes I love me some Atmas Impaler :P)

For early game purposes of surviving and squeezing out as much XP as possible I like to grab a REGROWTH PENDANT. This helps in multiple ways early game your job is to harass and farm like an animal. While getting Tantrum early you can lead your creeps into battle so you can proc your tantrum and start farming and harassing with it. I find regrowth pendant the only viable starting item if you want to play a successful Amumu.

3: Character Management

Applying Pressure

Applying pressure is very easy with amumu due to Tantrum. At rank 1 you must be hit 15 times before you can use tantrum which sounds like alot but is very easy to come by. Suggestions for gaining these 13 attacks. First lead your creeps into battle and if you have Regrowth Pendant as your starter item by the time the next wave comes you should be at full health and able to lead them into battle again getting one tantrum off per wave. I suggest trying to harass but at the same time make sure you are using tantrum to help farm gold as well. An easy way to harass is by running straight at your enemy and forcing him to run then come back and tantrum the enemy ranged minions. As you rank up the amount of attacks reduces making this skill darn near spammable. 

Jungling

I have recently been testing with jungling on Amumu and WOW its amazing. With this build once you get your dodge boots you are free to jungle like a mad man starting with the weak and eventually moving to more powerful spawns. When I decide to jungle I find using Teleport as one of my summoner spells amazing cause i can then teleport to good farming lanes. (Contrary to popular belief if you dodge it does count as an attack for Tantrum)

How to abuse bandage

Bandage Toss is AMAZING it can be used offensively and defensively. This is where it can become abuseive. I like to always lead my team into battle i bait the enemys to come towards the jungle(this works atleast 2-3 times a game due to peoples greed) having my team there to support me. An easy way to lead them into the jungle is by bandage tossing an enemy and getting hit to 3/4 - 1/2 life and then running making them think they are in control. At this point once you are out of creep range have your allies pounce. This is an amazing time to spring your ultimate and getting a kill and maybe a double kill but most importantly great amounts of xp for your team. At the same time if you are about to die after leading them into the forest you can either A: Bandage Toss a near by neutral to get away. B: Bandage Toss a creep in a lane to get away. or C: This is where it gets fun and very annoying for the enemy. If there is only 1 enemy left start running away and Bandage Toss back towards Your enemy stunning your enemy and now you are left with options again. A: If you feel like you can kill your opponent since you have the advantage of him being stunned now activate Despair and let that plus Sunfire cape and Tantrum if its up beat him to death or B: You are now free to run with little to no interruption from that champion.