A mix of tank and ap seems the best bet for Amumu, he is a very good tank, one of the best. His ult is incredible, he has a good stun, and he puts out wayyy too much aoe damage to be ignored in a teamfight. If you are up against a mostly physical team you can just stack sunfire capes and you are done. I find rushing a rod of ages is also pretty good for Amumu, it gives him his needed mana, more health, and ups the power of all his abilities.
hes a great tanky caster at least with easily full capabilities of full tank. i don't find him as useful being full tank tho honestly, and a lot of ppl don't time his ult properly. he has great...GREAT sync with a lot of other characters tho which basically makes him an always viable champ. flexible on his build with good lane and team fight presence
I started using the Tank mumu build because I've seen how weak an AP mumu build can be. Even against higher level opponents (I'm level 12 now, opponents were 19 and the other I don't remember) that used AP builds, they get killed much to quickly to be effective for the team. And considering the ulti causes you to get focused, health/armor is almost a necessity for mumu.
I guess I've never seen a truly good AP mumu build. You guys should get in a game with me