Tank engage and teamfight utility.
Durability and crowd control
After immobilizing an enemy champion gain defenses and later deal a burst of adaptive damage around you.
Increase your Tenacity and Slow Resistance for each summoner spell on cooldown. Immediately after casting a Summoner Spell, gain a larger burst of Tenacity and Slow Resistance for a moderate duration.
After 10 minutes gain 8 Armor and 8 Magic Resist and increase your total Armor and Magic Resist by 5%.
Every 8 monsters or enemy minions that die near you permanently increases your max health by 0.2%.
+65 health and +10 AP or 6 AD, adaptive
Gain 10% CDR when you reach level 10. Excess CDR becomes AP or AD, Adaptive.
Gain cooldown reduction and max cooldown reduction on your ultimate. This is increased further each time you use your ultimate, up to a maximum.
Aftershock is a great rune for entering a gank with [Q] providing you the resists to survive and a bonus burst of damage with the explosion.
Unflinching is a rune that synergizes well with most Junglers as smiting an enemy champion will grant you the Tenacity and Slow resist.
Conditioning on Amumu allows you to scale into the mid game as a massive tank alongside Aftershock and Overgrowth.
Overgrowth is likely the best rune in this slot primary due to Amumu's fast clear speeds and lack of major value from the other 2 options.
Transcendence is a great rune choice on Amumu as he likes to build all of the CDR tanks items, allowing him to go over cap and still make great use of the extra AP gained.
Amumu's [R] is the reason why you pick him. Taking the Ultimate Hat allows you to use your ult a bit more literally in small skirmishes and still have it up for big team fights.