So this is more or less the build I've settled on after playing Amumu for a few hundred games. For a long time I was using Ghost instead of Teleport, but I am starting to feel that Teleport lends itself to enhancing the game more overall, even if it means a couple of kills might slip away you are more able to be there when the team needs it, which also means you might be getting more assists anyway.

This is of course intended to be played as a lane Amumu. I have played around with Jungle Amumu some, but I just cannot manage to get it feeling right. Sometimes it works, but if you get thrown off your cycle or can't show up at ideal moments it can also be extremely ineffective. A couple missed bandage tosses on lane and you have probably thrown away most of the chance to pull ahead on a Jungle build.

I take Tantrum first for lane farming. It is absolutely vital that Tantrum gets used and abused to the maximum effect for last hitting. You are almost certainly the teams main tank and a fair bit of the CC, so you NEED to get the gold to get as far into the build as possible. Once you have Despair, this can be toggled on for brief moments to allow better lane farming. Just turn it on when things are low and toggle it again when most of the creeps are down, combining this with Tantrum.

As soon as possible, get the gold production items purchased and get back on lane. If you can, try to time this for when Teleport is up. As you transition into mid game, keep an eye on Dragon, with enough gear Amumu can probably solo it OK, but it doesn't hurt to bring some DPS. If you can snag a blue buff and your team doesn't need it then endless tears can help with both farming and lane pushing.

In general I try to initiate team fights with Bandage Toss / Curse of the Sad Mummy combo, followed by Despair and as many Tantrums as you can manage to get. The build I use is about crazy survive-ability so in most cases you are probably the first one in, and the last one out if it comes down to that. If you do need to die to allow more escapes or get another kill that is also fine, just leave the gold production items in their production form as long as possible so deaths won't weigh too heavily against the total build.

Towards very late endgame, I've found that adding on either a second Force of Nature or Warmogs can help depending on the lineup. Generally I'd abandon my Philosophers Stone in favor of that if I felt it was needed. Dual FoN can bring down even the most extreme AP teams, by just sitting in the middle of them with Tears / Tantrum / and burning from Sunfire.

If you do manage to actually max out the build, full elixer buffs can also help. I tend to prefer blue over red since Amumu's abilities actually scale fairly well, so my progression would be Blue -> Red -> Green -> Pink (if even needed).