If you want to stomp the pubsters with one of the best AP carries in the whole realm of League of Legends then this guide should help you.
You either want some magic pen, or all AP.
Case A: (With Magic Pen)
Grab all those greater marks for magic pen.
Put everything else into fixed AP.
Case B: (All AP)
Grab all red scaling AP runes.
Put 3 into blue scaling AP runes, and all the rest (blue and yellow) into fixed AP runes.
You want to have around +27 at 18 and +26 at 1.
I recommend case B, but I am also a huge troll.
Do that default AP carry mastery page. Otherwise you can fill it out however you feel it, it probably won't matter much in ARAM.
The good stuff. Start with the mana crystal and an 850 AP piece. Work your way into a Rod of Ages as quick as you can, then grab boots of sorcery and the death cap.
Once you have that strong base item set, work on Rylia's it is completely broken on Ziggs. Next step would be to grab Void staff, Zhonya's, and Abyssal. Situation-ally you could get morellonomicon if you are not a troll. Otherwise work on Liandry's for more pen, and lastly sell those boots to buy a Luden's echo.
Q Q + E at 1.
From this point on, prioritize Q and then E, and lastly W. Grab your R button whenever it is offered, and if you need a W earlier on to be a little more mobile feel free, just know you will hit a little less hard, not like it matters on this broken champion.
Mark and Flash.
Explanation: With your W and Flash you can bounce around the map tristana style. Mark is just so you can instantly auto attack scrubs who survive your onslaught of spells or during crucial engage and disengage situations to buy you some time. Especially nice with Zhonya's when the fight moves away from you.
You gotta throw the bombs where people aren't and let their human nature do the work for you. People will be sprinting into bombs, and remember you can toss them over creep waves and totally smash the AD carry. It's also important to never use R during the fight, just use it every time its off cooldown (maybe spend 5-15 seconds aiming it so it hits 3-5 people).
Pros / Cons
He murders carries, assassins, and pretty much everyone with less than 200 MR. He also murders buildings by simply dropping a spell and then attacking it. (That mastery helps here too).
The one con is that you aren't actually invincible, it just seems like it, so do not attempt to tank towers on this champion.
You are the team, just KS everyone except the AD carry. Any kill that goes to someone else besides you or the AD carry is grieving your game.
R - button, it launches an almost global (in ARAM) AoE nuke that will do upwards of 1200 magic damage pre-mitigation and it should be spammed as fast as you can get it up.