So, this is my first guide. I noticed a lack of Amumu laning guides. People say "Amumu is best in the jungle" that may be so, but he can also function as a laner. In this guide I will explain an Amumu build that has worked fairly well for me during my time playing LoL.

So my rune choices were:
Greater Mark of Resilience x9
Greater Seal of Resilience x9
Greater Glyph of Shielding x9
Greater Quintessence of Swiftness x3

The purpose for the Resilience Marks and Seals was to give Amumu some early game tankiness against auto attack laners since Amumu is a melee champion and is prone to harassing. The Glyph choice was selected for late game magic resist build up to help with his tankiness because of the rushed Aegis of the Legion Amumu will have a fair amount of magic resist by mid-game. The Swiftness choice was to help Amumu escape and chase. In my build I attempted to maximize movement speed with runes and masteries to maximize Amumu's chasing and escaping power.

Amumu's masteries are built similarly to a tank. Resistance and Hardiness give Amumu a further boost to his early game tankiness. The Tough Skin and Indomitable along with his Tantrum and high early defence makes minion damage very low early game and practically non-existent by level 9. Initiator helps with his movement speed for initiating and escaping. Veteran's Scars and Durability give Amumu some extra HP. Honor's Gaurd gives him some additional tankiness and Juggernaut gives him some extra HP and I personally use it as a substitute to Merc Treads.

The reason I chose Utility over the Offence tree was because of the movement speed given by Swiftness. Expanded Mind I chose because Amumu does tend to have some mana problems if you tend to be a spell spammer, but having extra mana is always nice. I chose Summoner's Insight because it covers both my Summoner Spell choices. Runic Affinity I chose because I had an extra point.

Ruby Crystal - Early hp, component of Aegis of the Legion. That along with decent start armor and magic resist give me a good start.
Null-Magic Mantle and Cloth Armor and Aegis of the Legion - I usually farm up and buy at once, teleporting back then using Teleport to come back to my lane if you need to get the items or happen to die just pick up the pieces then. The reason I do not get boots prior to Aegis is because of the movement speed masteries and runes allow me to keep up with most people.
Boots of Speed/Boots of Swiftness - I get this after Aegis allowing me to outrun most people at this stage of the game helps with kiting and the Aegis helps with farming harassing with autoattacks and tanking any damage I may receive

At this point its up to you how you want to build your item, should the opposing team be heavy in AD you may consider buying the Thornmail and its components first before buying the components of the Banshee's Veil. Now, Thornmail is not a necessity you can switch out Thornmail for Randuin's Omen or Frozen Heart depending on the situation.

After buying both items your armor should be around 200 and your magic resist around 150-160. At this point start investing in a Force of Nature to get to that nice 200+ range for tanking. (Along with that extra movement speed). After that get Ryalis's Crystal Scepter to put some damage behind your spells and add a slow for everything you do as well as adding some extra tankiness. Then get Oracles and the other consumables to boost the rest of your stats.

Max tantrum first it increases your tankiness by a lot early game.
Get Bandage Toss at level 2 so you have a stun. You want it for the stun, not the damage for now.
Get Despair at level 4 great damage because it does % damage based on max health. That along with Tantrum will be a very nice harass if you can chase down your opponent, which you most likely will be able to do with your amazing movement speed.
Get Curse of the Sad Mummy whenever possible.
Alternate between Tantrum and Despair.
Max Bandage Toss last.

I chose Teleport because it allows me to travel between lanes quickly allowing for ganks quick retreats and saving my tower if it is dying.

Flash I chose because it allows for escape and it helps with positioning with the Ult. I've tried other Summoner Spells for laning Amumu, it just does not work as well in my opinion.

Pros:
Early game survivability
Late game, great at tanking, nice DPS
Nice AP ratios
Fills as tank, support, and bruiser.

Cons:
Hard to kill people with high magic resist
High mana cost for spells so you have to be conservative with mana
Prone to ganks

Farming I found to be very easy with this build. Due to early tankiness he can walk up to minions and attack them without really worrying too much about damage. Tantrum is very good for farming, and should you have blue buff Despair and Tantrum combo can take out a group of minions very quickly. For this build because of early tankiness the laning phase if Amumu is being harassed he can easily activate Despair and use Tantrum to take out a chunk of HP from the opponent, the Bandage Toss and keep Despair on to do additional damage. Doing this every time an enemy attacks or get near should be able to keep them at bay and underfarmed or at the very least you should be able to keep up farm.

Amumu laning is based largely on his ability to survive the laning phase of the game and getting farm. After that he should be able to support the team in fights and be just as tanky if not tankier than a jungle Amumu and not needing to use Smite.

Thanks for reading. :)