Amumu is a great tank/jungler but no one realised is that amumu makes a good tankish ap carry due to his high ap ratios but always consult your team first and play what the team needs not what u want to play ^^

Pros*
-Tankish
-Good at solo top and jungling
-Has most advantage against top soloers like irelia / chogath / morde
- Despair works extremely well against tankish ad carries or tanks
-One of the best initiatives for teamfights

Cons*
-Not the best top solo, might want to give it to your team carries
-weak early game unless supported well

Jungling is optional for Amumu especially when you re going for ap build but if your team lacks for a jungle,go for it since Amumu is still a great jungler after all with Tantrum and Harden Skin u barely take any damage from minions so remember to take Flash and Smite

Most important job for Amumu is to start the fight by Bandage Toss in and use your ult so your team can finish off the ad / ap carries easily and it sounds easy but mistakes can still be made by missing a Bandage Toss so practice makes perfect!!

For summoner spells i take Flash and Ignite

Flashis a must spell for amumu cause it gives him the chance initiate the fight but flashing in and ult so u can give your team a chance to target the ap/ad carries and also amumu may be tankish but he still needs a spell for him to run away from ganks/etc

Ignite is a optional spell to take unless you're jungling but can prove very useful because it gives an early advantage over early kills

I always start with Doran's Ring for the extra health / ap / mana regen, next i head back to get boots and maybe another Doran's Ring, normally i end up with 3 x Doran's Ring and a Ionian Boots of Lucidity. Next i grab Hextech Revolver for the spell vamp which works really well with amumu spells and also a Catalyst The Protector which i will build into a Rod of Ages first then finish up my Will of the Ancients and by this time u are already pretty decent against anyone be it 1 on 1 or 1 on 2. At late game i would grab a Rabadon's Deathcap and a Rylai's Crystal Scepter for the extra cc and also i added Archangel's Staff cause if your team has an ap carry that might need blue buff more,u probably need some mana regen on your own.You also might need some tankish items like Thornmail or Force of Nature depending on the opposite team..BE FLEXIBLE!

For Runes i take Greater Mark of Magic Penetration for extra spell penetration, Greater Glyph of Magic Resist for some magic resist, Greater Seal of Ability Power for some early ability power and lastly Greater Quintessence of Ability Power for more penetration. Runes are each summoner's personal choice so to choose the most suitable ones , remember to experiment it!!

I would put a point into Tantrum for the extra reduce dmg incase of early gank or harass and put the next point into Bandage Toss. Remember these 2 spells are Amumu main damaging skills but i choose to max Bandage Toss first cause of the high ap ratio and also take Curse of the Sad Mummy whenever u can

Summoner spells:
Flash a must spell to take for amumu!
Ignite only if u're soloing top or going bottom
Smite only if u're jungling

Items:
Rabadon's Deathcap
Rylai's Crystal Scepter
Rod of Ages
Will of the Ancients
Ionian Boots of Lucidity
Archangel's Staff