Amumu is highly dependent on teammates, so try laning with your friends for maximum effectiveness.
Cooldown Reduction on Amumu is very strong, but it's often difficult to itemize for it. Grab the Crest of Insight whenever possible to gain Cooldown Reduction without sacrificing stats.
Despair is very effective against other tanks, so make sure you're in range of opponents with the highest Health.
Playing Against Amumu
Avoid bunching up with other allies when Amumu has his ultimate available.
Erratic movement, or hiding behind creep waves can make it difficult for Amumu to instigate a fight with Bandage Toss.
Amumu's Despair makes purchasing primarily Health items a risky proposition.
Cursed Touch debuffs an enemy's magic resistance which makes Amumu and his allies do more magic damage to affected targets.
Bandage Toss is Amumu's main initiation skill. He can use this as a gank/teamfight opener to get him into the fray or to provide mobility by dashing to minions/monsters to escape or simply to navigate over terrain.
During the time when an enemy champion is stunned from Bandage Toss, Amumu can basic attack first to apply Cursed Touch so that his and his allies' followup skills deal more damage.
Despair can be used to clear minion waves or jungle camps. Its % health damage allows Amumu to deal significant damage to all kinds of enemies even without any damage items.
Despair retains the same flat mana cost and cooldown no matter the skill's level. A high mana pool can allow him keep it on for extended periods of time.
Tantrum deals AoE damage around Amumu and passively provides a bit of physical damage reduction. Its cooldown is also decreased whenever he is basic attacked.
This is the Sad Mummy's bread and butter for clearing camps and dealing damage. Since he also takes reduced damage, he can easily tank monster and minion attacks. This in turn lowers the ability's cooldown, which allows it to be spammed.
Curse of the Sad Mummy is a great peeling and/or lockdown tool as it can affect multiple enemies in a very large radius.
Optimally, it should be used right after Bandage Toss to give enemies as little time as possible to react. It should also be used if multiple champions can be hit rather than just a single one.
Using the previously mentioned tactic, after the whole ensemble, Curse of the Sad Mummy can be added as a finisher or somewhere among the process to freeze targets with ease.
Amumu's skills become more potent with high ability power, yet his tankiness can let him stand in a fight long enough to finish off opponents. Fortify with armor and magic resistance items and ability power for offence.