Let's make some assumptions about jungle Amumu:
- requires blue for an efficient jungle start
- can gank at lvl 2
- builds defensive and AP items
- deals primarily magic damage
The first option (as you mentioned) is to steal Amumu's blue. Amumu relies heavily on tantrum to clear his jungle quickly, but this can quickly consume his mana. This prevents him from ganking and drastically slows his early jungle. (note: With jungle camps now providing health/mana, Amumu can still manage the jungle without blue. In times long past, stealing his blue completely destroyed his first jungle clear).
The second option is to ward/counter-gank. Bandage toss gives Amumu a strong lvl 2 gank. With map awareness and a key ward, the opposing jungler can counter-gank Amumu early before he has any defensive items.
Some junglers also have strong skills when dueling Amumu. Mundo, Shyvanna, and Lee Sin will typically win duels without outside help, and these champions excel at counter-jungling him as well. This can be especially true since Amumu tends to jungle with low health early on.
As the game progresses, Amumu's ganks will rely on hitting with bandage toss. Keeping wards in common gank locations and using minions for cover minimize his chances to succeed. Purchasing MR can help, and having cleanse or a QSS help immensely with escapes. Amumu is also susceptible to silences/stuns, so a team with high CC can prevent these. Also, a Banshee's Veil is very effective vs Amumu.
Janna deserves a special mention, as her ult will directly counter Amumu's if he doesn't kill her first.
tl;dr: Amumu is a tanky jungler and great initiator. He can be shutdown the same as other junglers via counter-jungling and early invades. His strength comes through mid to late game, but if he doesn't succeed early, he'll never get there. Strong early-game junglers who constantly steal his jungle are the best counter to him.
Best Answer
It definitely shouldn't be your first choice. The main Problem with Lichbane is that you require some AP for it to be efficient. Of course, if you are playing AP Amumu you could definitely say that it's a good and viable Item, however the viability of Full AP Amumu is rather questionable.
If you want to go for a bit of extra damage on Amumu or other low-scaling AP/Magic damage tanks in general like sejuani your best choices would be items like Liandrys Torment, Abyssal Scepter or sometimes even Zhonyas or Rylais Crystal Scepter. All of these items provide either enough Abilitypower or Magic penetration to increse your damageoutput by a good amount and they also Provide either Health, Armor or Magic resistance.
The main function of Amumu in a teamfight is bringing a lot of CC and a little bit of sustained damage while soaking up as much damage as possible. If you want to play an AP Assassin Amumu you will of course be able to oneshot ADCs (the scaling on your Q and Ult are pretty decent) but in general you should have enough burst in your team from mid or toplane. Amumus damage potential lies in low sustained damage, not in average burst damage.