Is Blumumu a thing?
Other than being fun to say, I'm wondering if it has any merit at all.
I don't imagine it being the most ideal build, but I'm curious if anyone has ever tried it, or at least thinks it may be worth trying.
I would imagine that most people already know what I mean, but basically "Blumumu" is working off of his mana pool for AP from 
Amumu's base mana pool is admittedly not very large, just shy of 1000 mana at level 18, which is not impressive.
His mana consumption is normally not much of a problem.
However, I believe he could go for some of the tanky mana items in order to get pretty unkillable while stacking up the tear by constantly using his W active.
My idea for a build.
I would probably start with Cinderhulk, and it doesn't have to be the Stalker's Blade version. Use whichever you prefer.
I don't think it would be as important to rush the tear because it shouldn't be hard to stack it up since he can constantly keep his W active once you have accumulated enough mana.
I think it may be better to go for the Righteous Glory as the second completed item, however it does give the least mana of all of the Eternity items. My thought is to try to be more healthy early on while you ramp up, and RG is the cheapest and offers the most health without stacking. The active on Righteous Glory and the GLP-800 would be good for preventing enemies from escaping your engage.
Iceborn offers the most mana of any item other than any of the tear items. It also offers armor and 20% CDR. The Spellblade proc also helps him prevent enemies from running away.
I honestly threw Morellonomicon because it offers 20% CDR, a good amount of mana, and a great amount of AP. At this point, you should be fairly tanky, and you may be turning more into a monster who just puts people down. This should put him over the top.
As for boots, he has no MR items so I figured MR might be good here. I really think that in most builds, boots are very conditional, so whatever you think works best.
But yeah, it may not be a good build AT AL, and don't jump all over me if it isn't.
Anyways,
good luck, have fun