9x flat armor marks - best in slot; does not need armor or magic pen since amumu will not be attacking nor does extra damage from his spells make him more useful. amumu is a tank; build him like one
9x dodge seals - my personal preference; better late game. if you want stronger early game, 9x flat armor seals work.
9x MR/lvl glyphs - best in slot; needed to tank better late game against AP carries
3x flat HP quints - needed for extra tankiness; doubt any other options are better here

1/13/16 masteries
1 point in offense for smite to help amumu with jungling
13 points in jungling for extra armor, MR, and dodge
16 points in utility for extra movement speed, which i feel is a must for jungling champions since they will be ganking a lot. cooldown reduction on flash is great as well.

i take smite and flash.
smite is a must on junglers. flash works well especially for his ult should you miss bandage toss. it also helps you to get away.

start off with despair, maxing out tantrum when possible and putting a point in bandage toss at level 4. max out despair next and bandage toss last. take ult whenever possible.

starting off with cloth armor and 5x health potions allows you to start off of at golem with a leash and you end with double buff at lvl 4. return to base and get boots and more potions and a ward or 2. gank or go through the jungle again. youre shooting for heart of gold, philo stone and either merc treads or dodge tabi. only get dodge tabi if the other team is stacked with ad. at this point, continuing warding and countering with oracles elixir. you have a plethora of items that you can go for. aegis is always viable. sunfire cape, randuins omen, or frozen heart against AD and abyssal scepter and force of nature against AP. buy items that will help your team and will work against theirs. always check your opponents' items.

1. Use your team's CV.
Always have your support player CV important parts of the map for you. This will help you change your jungling route should it not be safe for you or to counter jungle and take the opposing team's buffs when possible.
2. Use Amumu's passive.
Cursed touch allows you reduce an enemy's MR by auto attacking them. This effect stacks. Try to attack the enemy that your team will focus to help burst them down sooner. Your passive will help greatly.
3. Bandage toss has multiple uses.
Use it to initiate team fights with your ult. It helps you chase down enemies by stunning them long enough for your team to catch up. Use it on creeps, enemy regular creeps or neutral creeps for positioning purposes. This can allow you to get away or possible get to a spot sooner. An example of this would be attempting to get to a team fight at dragon sooner from bottom. Use bandage toss to go over the cliff onto dragon and help your team with a quick ult!
4. Timing for Curse of the Sad Mummy is important.
Using it too early or too late can be the difference between winning and losing team fights or ganks being successful or unsuccessful. Some champions require you burning off an ult on them to kill them, such as Corki, Kassadin, etc. Think wisely and time your ult correctly.
5. Don't forget your responsibilities as jungler.
Ward often. Keep track of neutral lizard and golems buffs, as well as dragon and Baron. Buy oracle's to clear wards of the opposing team. Gank often! Take over lanes when an ally dies or goes back to base to heal to prevent your lanes from getting pushed.