Basics

Jungle Path: Blue -> wolves -> Wraiths -> Golems -> Gank -> Red

Ganking: when ganking unless a clear shot is avalible refrain from engaging with your bandage as it's very difficult to hit a moving target with minions moving about.

Counter Jungling: Only counter jungle when you feel safe and have ward coverage and usally only in the mid-late game as amumu is very weak early. If you are invaded early (which you will be) simply take some of your team over to thier blue and take thier wolves too then go back to your wraiths and complete your jungle as normal.

Teamfights: Amumu is born to engage and since you will have around 3500 hp with this build you should rarely fear getting nuked. It's pretty standard to bandage toss in then position your self to lock up as many of the enemy team as possible. Hopefully if you communicated well with your team they should've followed up with thier aoe's and go in. If not, use flash to escape and rage at your team.


Reasons

I chose this build as most of the builds I see on mobafire don't take advantage of some of the greats items for amumu, they mostly build sup/tank only touching on ap slightly. However if you get Rod of ages and Rylias by the 25 min mark then you can dish out a ton of damage to a lot of people and have a large amount of hp to shrug off what they can deal back.

(side note)
If you want to, instead of selling your philostone buy an Elises Miracle it's not a bad item as it dissapears and becomes a passive after 3 levels.