Amumu has a very weak early game and is susceptible to invades. If you’re a strong 1v1 champion, try and invade him early to set him behind and delay his level 6 power spike.
Amumu needs to hit level 6 as fast as possible so he can gank with his Ultimate R. Try to gank as much as you can early on so your allies are stronger than his. This will make it harder for him to gank when it comes time to do so.
Like suggested, Amumu has a weak early game and will be looking to power farm. If he does show on the map, or if you’re stronger than him, you could invade his jungle and steal away some of his camps. This will delay his level 6 power spike and delay his Ultimate R which in turn, will also delay some of his ganks.
Amumu’s team fight presence is very strong. Do not group too closely as this will allow him to get a multiperson entangle with his Ultimate R. Do not fight inside the jungle either for the same reason.
Try and start a team fight as soon as the one before has ended. Fighting when Amumu isn’t there or when his Ultimate R is on cooldown is a surefire way of winning the next fight.
Spreading out the map and split pushing is your best bet as a 5v5 team fight against a team with Amumu doesn't usually end well. However, if you’re teammates are squishy and prone to get caught out, you’ll need to group.
Amumu’s level 6 power spike is incredibly strong as it allows him to start ganking thanks to his Ultimate R. You can delay his power spike by invading him early.
Once Amumu gets tons of cooldown reduction, his Ultimate R will be on a shorter cooldown. This allows him to gank more often and have more success when it comes to ganking.
Depending on the build, you’ll need to either invest in armour pen if you’re an AD champion, or magic pen if you’re an AP champion. If he is going damage, you might need some defensive items. As the game goes along, he will become stronger and stronger, and tankier and tankier.