A friend on mine asked me to make a guide for jangle mummy, so here it is.

Amumu works best tanky, therefore go 0/29/1.
You want the damage reduction to make up for the additional damage taken by not starting cloth+5, and all the defensive stats for doing your job as a tanky AP initiator. Bladed Armor is especially good on Amumu due to the nature of his Tantrum's passive, which enables him to take no damage from the weaker monsters and minions. Summoner's Insight also lets you get Flash that little bit earlier.

Always take Despair first when jungling.
ALWAYS.
I prioritize Tantrum first for fast clear and DPS when ganking, then Bandage Toss next to improve my burst and chasing power and Despair last to keep my DPS relevant late game. Take Curse of the Sad Mummy whenever you can, as it is Amumu's primary initiation skill.

Magic penetration marks: All of Amumu's damage is magical. Therefore, magic penetration is the only logical offensive option to take for marks.

Flat armor seals: A must on all junglers, also gives you some protection against the common AD bruiser top, jungle, and carry bot.

Flat magic resist glyphs: I've tried flat AP, flat CDR, CDR/level, flat armor and magic penetration. The end result is that flat MR is the best out of all of them, as it helps MASSIVELY when ganking mid after red on first time clear, as well as the occasional AP top. Flat MR, only way to go.

Spellvamp quintessences: Flat HP/5 offers less regen than spellvamp, tried and tested. It also helps you quite a bit when fighting someone else 1v1 and you have minions nearby. Gives the greatest synergy with AP mummy's playstyle.

Start off with a regrowth pendant, which will turn into a philo later on, and a health potion. After you clear and gank, you should have enough to buy boots + philo. Grab some mana potions if you can, as AP mummy is a very mana-hungry jungler. After your second clear and gank, you should have enough for merc treads and revolver. Now, onto Rylai's. Get Giant's Belt first if possible, then Blasting Wand, and Amp Tome last. After Rylai's, Will of the Ancients.

Now, this is the part where your build gets conditional. If they're leaning on the physical side, grab a glacial shroud to negate their damage. If the magic damage is too much to handle, negatron cloak. Whichever you got first, grab the other (Negatron > Glacial, Glacial > Negatron) and then complete the first item you got (Negatron > Abyssal, Glacial > Frozen Heart).

Your last item is purely up to you. If you feel that you're still too squishy, grab a Catalyst > RoA, or GA. If you need more damage, Deathcap, DFG and Morello's are all viable choices.

ONE THING TO KEEP IN MIND: Remember that the item order in this guide are purely situational. If your team needs you to get a Randuin's instead of a Frozen Heart, then rush Heart of Gold and ignore Philo. If Shurelya's is required, get it!

Alright, the fun part. As you are Amumu, the enemy team will most likely contemplate an invasion. Deal with it, jungle invasions against easily bullied junglers are a fact of life. Now that we've got that out of the way, you start blue. You can do wolves instead of golem first if your team is willing to lend you a hand, but normally you start at blue. Get a good leash, wait until it's smite-able, and then move onto wolves or wraiths, depending on whether you started wolves or not. Wraiths go by quickly, and now onto douoble golems. Get bandage toss, kill red with smite, and then go gank top or mid, using your own judgement.

Remember, Amumu's at his strongest when there's a ton of enemy minions around.
(Something to keep in mind when ganking)

This is tanky AP mummy. Not tankmummy, or OPmummy. Tanky Caster DPS, a rare role in LoL, is where this mummy falls.
/foreveralone
But really, have fun dominating.