I'm frequently building AP with Amumu, but only to give them a reason to attack me instead of my team.
I'm usually building: Cloth Armor, Heart of Gold, Sunfire Cape, Abyssal Scepter.
If I'm getting fed or they're ignoring me too much, I'll buy Zhonya's Hourglass or Rylai's Scepter.
But most of the time, I'm forced to buy either Aegis, Frozen Heart or Force of Nature instead.
It's defenitely a fun-build, as long as you still build very tanky.
But you NEED at least two of the following things:
1. jungling AND ganking succesfully
2. arranged team, that is built to support AP-amumu
3. stupid enemies
Maybe you can build him less tanky, than I do, but if you build him without tank items at all, you're just getting yourself killed.
Mejai's Soulstealer is already situational and questionable for most casters and it is defenitely a no-go for Amumu, because you're getting focused too easily with him.
Any halfway decent team will rape you, if you get Mejai's on Amumu, because you've just spent more than 1000 gold on something that neither offers a noticeable amount of AP, nor defensive stats.