Thank you Pallis123 for that enormous amount of advice. Anything unrelated to the actual game I have implemented in the guide (which is now on LoLKing). However, I chose not to change anything that is ingame such as the item build or masteries because my guide is just how I play it and should be unique from others (such as yours).

Again, thanks for your time!

First of all, I really love people playing unconventional but still some kind of viable picks! :D I discovered Amumu support a while ago myself as well and have played it quite often ever since, and I will try my best to give a detailed comment on your guide to help you with making it better (that's if you agree with my points, of course). I will split it down into commenting on your cheat sheet, different chapters and readability (coding / use of English) of your guide.

Cheat Sheet:

• You might replace the icons of Health Potions with the icons of Total Biscuits of Rejuvenation, because you took the cooky mastery;
• Your item section requires some improvement. It is not clear when you get which items (for example, I would get Sightstone as one of my first items, and not upgrade it into Ruby Sightstone yet, but in your cheat sheet you only mention Ruby Sightstone without other explanation. You could try to split the item section up in something like this: Starting Items --> First Back --> Core (/ Rush) --> Defensive --> Offensive --> Example Full Build. This is a lot easier to understand for people who read this guide and just use the cheat sheet for a fast idea of what they should build;
• You can not have 3 points in a skill by level 4, nor can you have 4 in one by level 5. I don't often use the skilling sequence you use (taking a first point in a skill at level 7), so I don't know the exact right order, but you should try that out in the game itself. I just know that this is impossible. :P

Pros & Cons:

• I don't agree with the fact that Amumu is weak in the laning phase. I even mainly pick him for a tanky champion with good laning presence. His E makes him very strong in trades because of the damage reduction it gives and being AoE. Q is a quite low cooldown stun and Leona E / Q in one skill, and his ult is just insane when combined with an AD carry. But that's just my opinion. Rest of this chapter looks fine.

Summoner Spells:

• There are more viable options than Flash and Exhaust. Ignite boosts your already rather high killing potential in lane, whereas Heal is in my opinion almost a necessary pickup when your AD carry doesn't take it;
• A little more in-depth explanation on the skills, such as when to take one over the other and more examples of what you can do with it, would be nice. Also, you should put some emphasis on the fact that you should always take Flash, and have to choose between the other 3.

Runes:

• Choices are perfectly fine. I would use them too, while maybe putting in some scaling magic resist glyphs if the enemy botlane has little to no magic damage. You can however explain it some more in-depth and give alternatives. For example, greater marks of magic penetration are also very strong for Amumu and boost his laning presence by a considerable amount.

Masteries:

• I don't like going 0/9/21 on tank supports, and will give you the mastery pages I normally run. If you do however want to run 0/9/21, there are still a few improvements to be made I think;
• The page I use in most of the cases, is 0/21/9. Legendary Guardian, Perseverance and Hardiness for example are incredibly strong and / or help your laning phase a lot more than gold income does. You still get the mana regeneration, summoner spell cooldown and cookies, meaning you have some of the important masteries from the utility tree.
• 0/16/14 is the golden midway between 0/9/21 and 0/21/9. This way you forego Legendary Guardian and Tenacity for more gold income and starting gold, allowing for you to scale better and get your items faster.
• The last possibility is 9/21/0. As you mention yourself, Amumu should be played as a kill lane support, and more ability power and increased damage for teammates helps a lot with doing damage. Hell, for a midway between everything if you want you can even go 9/12/9.
• If you want to go 0/9/21, you MUST take points in Summoner's Insight. I would either remove 2 points from Fleet of Foot and the 1 from Scout / Phasewalker, or take 3 out of Meditate. Depends on if you think you need the mana regeneration or not. Rest of your mastery page is fine.

Abilities:

• I don't agree with maxing Bandage Toss. It is a one-point wonder, and however it is true that the cooldown goes down and the damage goes up, Tantrum scales far better with levels. It is your bread and butter for trading and gets more damage, more damage reduction and lower cooldown. I would go this way: Q-E-W-E-E-R and from there on max E, then Q, then W.
• Just take a point in W somewhere early instead of level 7, as just turning it on early both gives some damage and doesn't cost that much mana if you remember to toggle it off.

Items:

• I don't agree with your core items. While Talisman of Ascension in theory can work and actually is good, I think Face of the Mountain is a better option. With your build you go over the CDR cap, so Talisman giving more CDR than Face of the Mountain is not a good argument. The 500 health is a lot better than mana regeneration. The only thing Talisman has over Face of the Mountain is the active.
• I wouldn't get cooldown reduction boots. Amumu is not that CDR reliant, and most support items already give you more than enough. You are better off with Ninja Tabi or Mercury Treads for more tankiness, or Sorcerer's Shoes for more damage. Even Boots of Mobility are an option, but Boots of Lucidity not in my opinion.
• There are a lot of items that you could mention, assuming you don't get the same build every single game. Abyssal Scepter, Frozen Heart, Sunfire Cape, Banshee's Veil, to name a few. I will let you figure out yourself which items are worth mentioning, but I guarantee you that there are more options out there than just the ones you listed.

Gameplay:

• Your tips are fine, but I guess that instead of putting in random tips (however they are absolutely good!), you could also split it up in chapters like Laning Phase / Mid game / Teamfight / General supporting. That gives it a little bit more structure and makes it easier to understand.

Readability:

• You seem to have a decent understanding of how columns work, and assuming that I would like to ask you to space out everything a little more. It's all so close together (for example in your items section), and that makes it hard to read. You could try to add an enter more between every item + explanation, and maybe also add a blank nextcol with a width of ~ 20 in between every icon and item to make it look a little neater. Same goes for Summoner Spells, where I think your images are a little too big which distracts me. Your abilities seem perfectly coded to me. :) Overall, you seem to have a basic grasp of what the coding works, but try to use it more! Coding is very important and one of the biggest parts of getting people to like your guide is the aesthetic appealing.
• Your English seems fine in most places. There are some stylish thingies that I would do differently, but that's all. If you want me to be really nitpicky and review the entire English, I would be glad to do so. But as I'm not sure if you want me to, I won't do it (yet).

All in all, I'd say the fundamentals of the guide are good, but it still requires some work before being a good guide. I will not vote, but if you decide to update things and take my comment into consideration, feel free to PM me and I will have another look (and maybe upvote).
This whole wall of text might sound a bit negative, but I'm very nitpicky, so don't take it personal or think your guide is bad, because it is not.