This is not the worst ez build ever. However, it leaves much to be desired.

For items:
I consider Lich Bane a core item on AP ezrael. It's passive procs on using his Q and is consumed when his Q hits, so it is utilized much like trinity force and sheen on AD ezrael. It essentially gives his Q a 1.2 AP ratio, adding the proc to it's natural .2 AP ratio. Seeing as your Q is the lowest cooldown spell you have, this significantly increases your damage output, and is your only legitimate farming skill. Additionally, it's mana synergizes with Archangel's Staff and it's movement speed is extremely useful on a skillshot champion that requires good positioning in order to land hits. That 8% movespeed, while not as large a boost, will substitute the movespeed you lose from not doing AD Ezreal (phantom dancer).

Also, Lich Bane supplies magic resistance that, while not equal to Banshee's Veil, would allow you to swap your veil for something both offensive and defensive. My pick is Rod of ages, as it gives survivability through it's hp, mana that synergizes with Archangel's Staff and is an asset on its own, and AP that can be utilized with 3 skills (4 if you have lich bane). Also, Rod of Ages and the catalyst used in its recipe have a passive that helps with sustain, allowing you to stay in lane for much longer.

Keeping this in mind, I tend to forgo a doran's ring and its earlier advantages for a Meki Pendant, which is good for sustain and can be turned into a tear, which is very advantageous considering that ez's skills are very spammable. Another good early item is a sapphire crystal, as it can be built into a sheen, tear, or catalyst, depending on what item you want first. Finally, a good starting item is boots, as ez is, as I mentioned earlier, very dependent on positioning.

On the subject of boots, I recommend swapping your Sorcerer's Shoes for Ionian Boots of Lucidity (AKA cdr boots). With lichbane, your q will still deal the 1.2 AP ratio damage as physical damage, so the magpen will not apply to that. Also, your build has no cdr items and despite the internal cdr on Ez's Q, he still has moderate cooldown lengths. Its natural cdr synergizes a bit with your Q, giving you pretty decent cdr. If you really need magpen, just go for a voidstaff.

You were right in going for deathcap, as it will synergize with Archangel's passive. Every time you gain AP from Archangel's Staff, it will be enhanced by Rabadon's Passive.

The above changes would give you a massive mana pool for a spamming champion, movespeed for a skillshot champion, overall better burst (which is the point of going ap over ad on ez), a decent earlygame (which is usually the hard part for ap ez - just go for sheen and/or tear, as ad ez usually does that too for trinity force and manamune), and a legitimate farming ability.

I don't want any 'this is the worst ez build ever'. If you do this, please show how it could be improved. Thanks!