Play by the two-bandage rule: walk up or Flash-R if you can, use the first Q for access, and keep the second for the panic Flash, dash, or late diver instead of burning both on entry
Better Amumu fights start in walk-up ult range, especially from the lip of pit or a ramp, where you can lock two bodies first and bandage the escape after they panic
On first clear, favor lanes where the target has to run a straight line and skip early plays that only work if blind max-range bandage lands; recent data backs that up by showing Amumu is less early-gank dependent than several similar engage junglers
Start by shrinking the fight instead of forcing it. Stand where the enemy has to walk through you, usually the lip of pit, a ramp, or the corridor behind your front line, and make them answer your body before they answer your Q. If they step too close, walk into ult range first and use bandage after the panic move; if they do not, keep threatening the choke and let them be the ones who lose patience. In even or losing games, this often means hovering your carry and turning on the diver rather than trying to reach the backline yourself. The key is to create a short path to contact before you ever throw the skillshot.
Amumu's basic attacks Curse his enemies, causing them to take bonus true damage from incoming magic damage.
Amumu tosses a sticky bandage at a target, stunning and damaging the target while he pulls himself to them.
Overcome by anguish, nearby enemies lose a percentage of their maximum Health each second and have their Curses refreshed.
Permanently reduces the physical damage Amumu would take. Amumu can unleash his rage, dealing damage to surrounding enemies. Each time Amumu is hit, the cooldown on Tantrum is reduced.
Amumu entangles surrounding enemy units in bandages, applying his Curse, damaging and stunning them.