Ok, so I am not claiming to be a know it all on Amumu. He was just the first champion I ever played and I really wanted to share my two sense on him. I know that he's a traditional tank, However i believe he still does exceptionally well as a tanky mage.

Pros
- High CC- High CC
- Suprising damage with decent ap scaling
- Remains tanky even when building carry

Cons
- Dependent on blue
- Counter Jungle cripples him
- Long CD Ult

I take 9/21/0 grabbing the AP, CDR, and Magic Penetration in offense, while grabbing defense masteries for an easier jungle focusing on health, CDR/lvl, and increased movement speed for ganking potential. This mastery setup is designed to give u tankiness throughout the game while still providing the CDR for ult and wrap.

My runes focus more on damage than the tank part of amumu. I focus for higher dps on a gank to help secure the kill. Also the AP glyphs and quints give me a decent clearing speed and the armor gives a slightly higher survivabiliy.

Personally i like to run Flash because it has amazing synergy with his Q and ult. Flash can be easily used for an initiation, escape through walls and other obstacles and used to catch up to a fleeing opponent in order to Q him.

Ghost is also a viable choice but with movement speed 3 boots flash is just a better pick imo

There are a couple of variations to this build you can go.

One variation is with Lich Bane. Lich Bane adds 100% of your ability power to your next auto attack after an ability is used, therefore it adds a nice burst after you successfully wrap a target and also after a tantrum.

Another nice item to pick up on him is Zhonya's Hourglass if you find you are vsing a fairly ad heavy team. That active works nice after a flash + ult initiation and can even cause a couple of ults to be mistakenly blew for no profit.

Shurelya's Battlesong can be built from your Philo stone and provides a nice active and cdr for initiation, running or chasing.

Have your mid damage wolves as you take them. You do this so that you can grab wolves - blue - wraiths and back to wolves. This is the fastest way to reach 3 and will give you the best success when trying to gank an enemy in lane. I strongly recommend a leash so that you have more jungle time and gank time and dont need to B early because you lost too much health at blue.

You are a jungler. This means that your main job is to assist lanes that are having a troublesome time. This does not mean gank and take the kill every gank because in the end that will just end up hurting your team as a whole. Amumu's gank sequence revolves around letting your lane know your ready to gank and using his wrap to disrupt the enemy AFTER they have decided to run. If you wrap to early and are pre 6 then there is no longer a cc because of bandage toss' long CD. If you happen to be level 6 make sure that your ult must be used to secure the kill before you use it. Amumu's ult is a game changer and blowing it on a gank that could have been secured easier or blowing it on a gank then still not getting the kill is a big loss for your team as a whole.

Amumu should always look for a gank after he hits lvl 3. By then a lane will more than likely be pushed up and you can coordinate your cc in order to score first blood.

In a team fight people aren't going to expect your damage. The popular belief is to not attack tanks. When the enemy team does this you punish them with you high scaling and very nice AoE. If you see a good oppurtunity let your team know activate you W and then bandage toss to a carry, ult, and tantrum. This will more than likely atleast make the carry back off while your team comes in to clean up the devastation that you just caused. Proceed to help your team mates with Bandage Toss, Rylai's slow, etc.

Warding their blue can provide an easy chance to smite steal it. Grab a ward and slap it down around the 6/11 minute mark and you should be rewarded nicely with a free no work blue.

When being chased entering a bush then bandage tossing to the target and running the opposite direction can really save your ***. Not only will you have that brief stun but also the slow from rylais applied to said target.

If getting chased and your near baron or dragon, you can bandage toss through the wall to get away, if you are followed with flash you can then flash back through the wall and proceed to trololol as you run away.

With Tanky Mage, You aren't just another tank that sits there and just absorbs damage. You can absorb damage and deal out a great deal with this build. You will stay tanky enough to survive focus while still dishing out enough damage to be able to be considered a serious threat. Like i said before, the other team will not expect your damage output therefore they will do the norm and not focus you in a team fight.

I am still continuing to update this guide and make it looks better and adding more information.
So please don't down vote.... at least right away because over the next few days it should change drastically.