The creator of the guide said 2nd archangels and third archangels or a void staff, so read the stuff inside the build (A voids staff was include but it was assumed that if they don't get much mr then you don't get a void staff... also your assessment is off on the amount of damage done. Also the more mr they get the less deadly they are to the rest of your team... unless your team is all ap for some reason.... plus the lich bane makes up for those who get mr... unless most of them get mr it's poitnless.

You should probably include a void staff, once they start getting magic resist, you're going to be hitting for around 10% of their health instead of 20-50%. And as previously stated, building Rylai's is actually good because it allows you to chase with your team. Other than the triple Archangel's this guide is pretty decent.

This is how I used to level up his abilities, but then I realized his W helps alot in situations his other spells don't for early game. I say that anyone can say stack archanges and you get the desired effect... but that's not a well polished item set for AP kog. Your kog you say doens't have enough of a mana pool to continually ult even without blue buff if you don't have a chalice. I say that is false for my kog can do that and it has around 800 less mana than your kog.
Also you are not maximising the cost/benfit health:mana:damage:speed ratio. By this I mean if your oppoent has boots of switness they can outrun you unles you ghost, this is where a rylai's would come in handy, so if the oppoent jumped you, you could have more survivability. I do not believe lich anes passivies stack nor do archangels thus the 2nd lich bane is just for ap and movement speed I assume and the archangels for mana and ap. In other words this build is ok, but I don't believe it does AP kog justice... although I wont downvote it for it is viable... although not my preference.
(The only time I ever run out of mana is when I use other abilities on top of my ult, and then jsust a quick 5 second break and I'm good to startup again if I have blue, if not a quick 10 second break and I have enough for a real teamfight and not artillery spam. If no blue and fully loaded on mana then I have enough for around 20-30 shots dependant on how fast I fire... for sometimes there's nothing to shoot at. Basically you ahve 4k aman... at 200 mana per ult that's enough for 20 shots... then your mana regen per 5... each ult taking a second so.. 20 seocnds times 105 this another 2 ults minimum without chalice or blue buff...

I can understand the summoner spells and bravo on the ranked play... for I strongly agree champs like nocturne and whatknought can be annoying (althouh since nocturne makes you blind you might as well jsut looka t yourself... for that's proably where he's going... then jsut ult on yourself even if he's not there.... and then if he shows nuke him down asap or run for your life, yet I disagree that Xerath is a worse version of kog... his ult has a way logner cooldown. The chalice is wise early game. The reason I don't do it is I get a rod of ages for survivability, mana, and ap, and those stacks are needed sooner than later.

To add to the annoying early game champs I would have to say orianna... you rarely see her, but with her ult she has enough range to fight you until your level 11... even if you are great at skillshots... (especially since she can pull you closer).

To add to late game hinderances... competnet twitches... for your usually behind the battlelines... thus I say kog needs a bodygaurd for if you don't see twitch coming... your screwed more than if it were any other champ... Of course this is assume twitch built properly... (I even find it hard when with my build which has more ap and health than this 1.... damn those lifesteal champs!). AKA don't let lifsteal champs near you... unless they're squishy... for your lich bane and all your abilities will nuke them... lee sin is annoying

When it comes down to it... I find no champ can fight ap kog unless they have some serious lifesteal and attack speed... haven't found a champ with spell vamp able to outdo ap kog.

wow thank you :)

Since you requested in the introduction for correction of any spelling or grammar mistakes I'll post what I found whilst passively over-viewing below. May of missed something keep in mind.

1st!