I've been playing as Amumu as an AP/Tank and I'll sometimes play him in Jungle. I was just playing in a bots game experimenting different items and here's what I finished with:
Athene's Unholy Grail
Banshee's Veil
Mercury Treads
Sunfire Cape
Abyssal Scepter
Ryali's Crystal Scepter

*Note: This is just what I was trying out. This is not what I'm calling my official build, so please no comments saying, "Your build sucks" but any advice would be appreciated.*

When I asked my teammates for their opinion on whether or not it looked good, one of them said to drop Sunfire Cape for something else like Guardian Angel.
I thought Sunfire Cape would be a good item to always have on Amumu seeing as how he gets close to enemies a lot and has lots of AoE abilities, but again, I'm just learning him so I don't know if it should be replaced.
Should I keep it or drop it late game?

seems ok most games you u never get full builds though not on a tank. i been playing amumu since level 15 though i stoped playing him some 250 games ago due always being banned but now i am back playing amumu and trying ap builds

wow your level 30 with 26 wins think need stop playing bots and play real games they much more challenging then bots. i have 476 wins 578 losts in nomal games 276 bot wins and 18 bot losts.

Too much ap. People will rage. Grab a Fon/ Randumin omen for Rylais or abysmal.

Build more tank less ap
IE- why int he world would you ditch sufire cape?!!!?

Ok, thanks! Is Athene's too much, though? I find it helpful at least as a late game item. I can pass up Rylai's, but I kinda like Abyssal Scpeter because it gives MR and AP and it's aura is really helpful, too.

believe it or not i think going full tank with the right items actually does more damage, but i like this build personally: philo,ROA,sunfire cap,either shurelias,banshees,FON,frozen heart, thornmail. the last items depend on enemy team, iv seen people 3v1 with this amumu build because sunfire plus thornmail is rediculus vs ads and FON,shurelias can completely zone just about any ap.