Hi, Welcome to my guide! I started playing League of Legends 2 years ago. My first character ever was Amumu, over the past 2 years of playing, I have played Amumu serveral ways. Tanky, AD, AP, Support, Top Lane, every way you can play him. I have come to a conclusion that AP Amumu is most powerful and useful! So here is my guide, ENJOY!

Pros
[*] Strong burst
[*] Good team fight engage
[*] Able to take on multiple enemys
[*] Clears minion waves fast
[*] Unexpected by enemy team

Cons
[*] You gotta be good at skill shots
[*] You have to engage, and if you forget your ult your team is screwed
[*] Drops tank role if you don't have one

As you can see this rune page doesn't revolve around the build itself. The runes are typically only for early game, helping you jungle easier. Also armor marks and seals make up for some of the lost tankiness you could build. MR obvious reason, and Movement speed Quints are good for getting around or chasing an enemy.

Jungling with Amumu is quick and easy. Start at blue, and move to wolves after, after that move on to wraith camp, and then golems. Your smite should be ready after Golems. You can either gank top or go home and pick up Philosophers stone. After you buy those, go kill wraiths again, and then red. If you can gank a lane at this time go for it.

Amumu is also good for covering lanes when your team mate goes to buy or heal up. So when you get the chance pick up all the farm you can!

Now on to the items! There's a lot of ways to build items, it all really depends on if your ganks are successful! If your not getting kills then there's alternate ways to build.
Also it depends if the other team has heavy AP or AD. I'll make 3 different guides to implement games.

Heavy AD team;
regrowth pendant -> philosophers stone -> boots -> health crystal -> heart of gold -> sorcerors boots -> chain vest -> Needlessly Large Rod -> zhonyas hourglass -> negatron cloak -> blasting wand -> abyssal scepter -> blasting wand -> voidstaff -> Needlessly Large Rod -> blasting wand ->rabadons death cap -> sell philosophers stone get giants belt -> sell heart of gold get blasting wand -> rylaicrystalscepter

Heavy AP Team;
same build, just switch zhonyas hourglass and abyssals around. same build after that.

So why the choice of items?
philo stone and heart of gold for GP10s, sorceror boots for magic pen, void staff for magic pen, abyssal works really well with Amumus passive and lowers MR on other team, rabadons for crazy damage, zhonyas for tankiness and more damage, and rylais for hp.

Why Zhonyas?
when you're about to die zhonyas and you have time to look for a champ and get your bandage toss ready (bandage toss scales late game with all the magic pen you have)

Despair -> Tantrum -> bandage toss (for ganks) -> max out tantrum -> get ult -> max out despair, max out bandage! simple! Tantrum to clear jungle fast, despair to help with that also, and bandage for chasing and ganking!

Flash + ult = good initiation and saves your bandage for more dmg
Smite is for jungling!

Amumu and Orianna make a kick *** team!

Full tank masteries and some AP caster masteries make Amumu kick ***.

This build might seem absurd, but you must try it before you say anything!
Cheers!