This build is based off Jungle Amumu: A Guide to Depressing your Enemies with changes, I like to give credit where credit is due.

I prefer to use "Ghost", "Boots of Speed" "Void Staff", and maxing out "Despair" versus the other build. This will make your Amumu jungle faster, more offensive and really helpful in team fights.

Now to my guide, I start with the standard Cloth Armor and 5 Health Pods. I start at Blue Buff and work my way to wolfs >> Wraiths >> Golems >> Red Buff >> Wraiths >> Wolfs (By this time you should have enough to buy boots of speed outright.

After you get your boots of speed you can start ganking to gain gold and push lanes. At this time don't push kills just push lanes, not worth getting killed over little gold. Continue to Jungle until you have enough to buy you Sunfire Cap this will greatly improve your offensive output. At this time you can push more kills as you should have your ULT.

I like to max my "Despair" because added with the Sunfire Cap you have a strong early game presence and the ability to change team fights.