Hello, this is my first guide and it will be rather short. I have had recent success with Amumu in the jungle once I deviated from the top Amumu guide with my own modifications. Four wins out of five and very good scores regardless of the match outcome. I found that provided there is no other dedicated tank having a decent amount of CC, the extra health I build early helps tremendously.
I found this build to work very well in a game I played with a team set up as follows:
Top: Poppy (though I am sure any competent solo top would do fine)
Middle: Galio
Bottom: Nunu, Caitlyn
Jungle: Amumu
This team set up allowed for devastating combos provided our team took our time once the team fights began and had a preset strategy; Amumu, initiate and once Galio and Nunu are close or within flash range use Ultimate. Then Nunu uses ultimate and Galio as well. Ideally at the same time, just as long as Nunu is not channeling alone at the start in order to not get disrupted. With all that happening Poppy runs behind the team to curtail any runners while Caitlyn does what she does best, mess up whoever survives. This resulted in Nunu getting a number of kills despite being fairly underfarmed early. We didn't win every team fight but this strategy forced them to stay separate which Poppy, Ammumu and Caitlyn took advantage of.

Standard jungle route, making sure to have appropriate jungling masteries (check above). Blue, wolves, wraiths, red or golems --> gank.

Despair, Tantrum then Bandage Toss. After that your priority is your ultimate, tantrum, despair/bandage toss. Despair first if your opponents are more melee and bandage toss if they are more ranged. Since you are playing an offtank the stun is more useful and leveling it lowers the cd.

I find this build suits Amumu as you are able to initiate a little more safely, by all means swap out lich bane for Guardian Angel if you get that far. Amumu is great for his stun, getting in the middle of the enemy and doing decent AoE damage and his amazing kill securing ultimate. Try to stay within range of the furthest enemy carry (or whoever the main target of your teammates is and use your ult to stop them from running away. I find that most teams know not to physically attack Amumu, which is also why I build more magic resist that most on him. Frozen Heart and Lich Bane are Optional and sometimes I build Abyssal Scepter or Guardian Angel. But usually I am never low on health at this point in the game so I don't really recall much. If I ever do the last items are very subjective to the enemy team, and what my team is lacking. Sometimes I even build Rylai's. Wish you all luck and good games, thanks.