Try to not get stuck on bottom. Amumu with this much health just has to try not to run out of mana. For boots I pick either ninja tabi if there team is heavy AD or Mercury's Threads if they are heavy AP. The best way to play this is go to the battle and tank use Q Bandage Toss and R Curse Of The Sad Mummy to stun and keep enemies on your tower using the tower helps you get fed fast. I prefer to have a mage like Ryze or a melee like Jax up with me I can keep enemies busy while these high damage classes take them out one by one.

I always play defensive capping 3 towers and just holding them. If their team is mostly AD I like to grad some Thornmail just to laugh as they kill themselves while attacking me. If they are heavy AP grab Force of Nature or Guardian Angel and watch them throw everything they have at you then run from you.

Proslives forever
AoE damage
while E is ready you get reduction to physical damage
Lots of fun to just watch 2 or 3 enemies try to take your tower as you get a triple kill because they can't cap because of your AOE hitting them all
Never run away from a group run to them watch them run

Conslow damage
hard to get to range if you miss bandage toss