I recently ran a AP Ezreal like this:

Boots+3p
Tear of Goddess
Ionian Boots
Sheen>LB
Manamune>Muramana
DFG
Sell Ionian and get Sorcerer Boots

By this time his damage was insane, the combo E>DFG>W>Q>R is strong, and can also go with a Liandry's Torment and a Iceborn Gauntler (haven't tried though) or a Void Staff. Try it if you want it works nice. Still I be posting any new things i see on this build. Although Early game need to play very passive until you get LB.

AP Ezreal always needs to play passively early game, nothing new. He is one of those champions that try to survive the lane and not win it.

Other than that, I think that Muramana on AP Ezreal is anti-synergetic, since muramana gives AD and not AP.

But remember the passive of Maramune. "Your single target spells and attacks consume 3% of your current mana to deal twice the amount as magic damage".

Don't know if i'm not interpreting good the sentence but i think it says that it doubles your magic damage at the cost of 3% of your mana. Please correct me if i'm wrong.

But Q is a physical ability why it says magic damage? Thats whats confusing me. I get your point though with the Q.

not really a good idea manamune's an ad item for one and yes the double damage is nice but ap ezreal's scaling on his q sucks compared to ad u could easily get twice the damage on it from just building flat ad and double that with manamune.
His e is mainly for escapes and is the only skill that's actually decent with that manamune proc but if ur using that ur basically wasting 2k of gold on an item that gives 0 ap and won't boost it up and u've gotta get it early since u need to get the 750mana stacks asap.

I think you should go archangels, not manamune...

Muramana is for AD casters, not AP

Eh? Muramana makes AP Ezreal actually have fighting capabilities, instead of poke and burst only.