For masteries I've found 9/0/21 to work best for the increased movement speed and cooldown reduction. The 9 points into offense nets you 4% Cooldown reduction and 10% more magic penetration while the points in utiliy get you an extra 6% cooldown reduction more movement speed and some bonus experience to level up faster.

Utility Tree:

Good Hands 1/3 --- In case you die, it comes in handy more then the others.
Expanded Mind 3/3 --- This helps with mana which can be a problem, especially early game.
Swiftness 4/4 --- Movement speed to keep up with enemies and move to bases.
Meditation 3/3 --- Helps with the mana problem again.
Transmutation 2/3 --- Spell vamp is never a bad thing, land some aoe hits and collect health.
Awareness 4/4 --- Bonus experience... never a bad thing.
Intelligence 3/3 --- Cooldown Reduction is key for more ultimates and bandage tosses
Mastermind 1/1 --- Get to use ghost and ignite more often.

Offense Tree:

Summoner's Wrath 1/1 --- More move speed on ghost and bonus Ability Power when Ignite is on Cooldown.
Mental Force 3/4 --- More Ability Power to dish out some more damage.
Sorcery 4/4 --- More Cooldown Reduction
Arcane Knowledge 1/1 --- 10% addition magic penetration for more damage.

I've been playing Amumu quite a bit recently in Dominion and I've seen several different builds. I decided to share my own personal build here. Keep in mind this is my first guide. Using this build I've been able to stand toe to toe with any other champion 1v1 or 1v2 and still come out on top.

Pros:

Tanky
AoE Crowd Control
Considerable Damage with bandage toss
Stun on bandage toss
Can solo defend a base against 2 or 3 people

Cons:

Reliant on team
Relies highly on landing bandage toss
No escapes

The rune selection gives you some extra survivability, mobility and some added damage.

x9 --- I choose these marks to help get some damage more damage in on enemies.

x9 --- I choose these seals for the extra armor early game for survivability.

x9 --- I run with Ability Power per level glyphs for the extra damage.

x3 --- I go with movement speed quints so I can keep with my enemies as well as move around the map cover bases faster.

As far as the skills go a lot people on Amumu choose to max Tantrum first. While tantrum is nice to have I find in Dominion that 3 points into is enough for the early game while massive damage bandage toss can do makes it better for mid game. So off the start I grab a point into each and then put the next 2 into tantrum. Next I grab the ult and max bandage toss.

So it goes Bandage Toss x1 -> Despair x1 -> Tantrum x3 -> Curse of the Sad Mummy -> Bandage Toss x4 -> Curse of the Sad Mummy -> Tantrum x2 -> Despair x2 -> Curse of the Sad Mummy -> Despair x2

As far summoner spells go a couple different ones work with Amumu.

I normally go with Ignite and Ghost for the movement speed and the extra damage and reduced healing

Other good summoner spells would be Exhaust and Flash.

Now I understand my build doesn't have a lot of AP items and is mostly tanky but hear me out.

I've noticed Amumu doesn't need excessive Ability Power to do sufficient damage. With the Cooldown Reduction from masteries and the boots Bandage Toss has a 6 second cooldown at max rank making it usable fairly often and even with this build it normally takes chunks of peoples health. With the movement speed boosts they can't really out run especially considering the stun bandage toss applies.

So I build tanky so I can initiate more for the team onto bases as well as defend them longer

I normally start out with Boots and Blasting Wand.

Next time I back I upgrade my boots into Boots of Lucidity or if I have enough gold a Giant's Belt. Grabbing which ever I don't get the next time.

Following this its best to look at what kind of damage you are taking in fights, rather its magic or phyiscal.

If the damage is mostly physical I get Chain Vest next and upgrade it to Sunfire Aegis

If the damage is mostly magical I get Negatron Cloak and upgrade into Abyssal Mask.

Both of these items add a significant boost to damage and survivability so I normally end up getting both.

Next again depends on the enemy teams damage. Most games by this point AD carries are the biggest problems which is why I normally build Thornmail. However if the enemy team is still doing significant magical damage I get Negatron Cloak and start upgrading it into Odyn's Veil.

Most games the enemy team is balanced, however if you arn't taking much of one kind of damage, like the enemy team is 5 AD champs or 4 AD and 1 AP or 5 AP or 5 AP 1 AD, I would then just build which ever of the above I needed and would go into Rylai's Crystal Scepter. The scepter is also a good idea if the enemy team is dishing out a lot of true damage since neither armor or magic resist helps against this. Champions to watch for on this include Teemo, Kayle, Vayne, and Cho'Gath.

As far as playing Amumu I'm all about the bandage tosses. Knowing when to go for it in team fights and being able to land them are key aspects of playing him. When you initiate a team fight with Amumu or are coming into the middle of one turn on despair and try to land a bandage toss on their heaviest damage dealer and then pop tantrum. Next try to make sure you can get multiple people in your ult before you pop it. Then just spam toss and tantrum til everyone goes down.

Using this build I've been able to stand toe to toe and come out on top against every champion out there.. Granted some were harder then others to beat...

Thanks for reading and good luck.