Lux's early Game is really bad with Cooldowns, Thats Why I get CD Reduction with CD boots and Evil Tomb.
You will need to get A Lot of Mana pots becuase Lux is a Mana Eater if You do What I do when I spam the **** outa her Abilitys. Also Take Caritly it Will help you out alot When you are Harassing the other team with your Abitilys. at Lvl 9 You need to max out your Q First for the CD Reduction, The reason you do this is because How I play Lux I just Hit someone wiht my Q and then Spam my W and R when you can, Also try Hiting them wiht your Q before firing your R so you don't miss.

Mejai's Soulstealer:
I use this for the CDR, What it does is that eaach time you get a Kill or an Assist, this Stacks with AP Kill is 2 Stacks and Assist is 1 Each Stackes gives you 8 AP and at 20 Stacks gives you 160 AP + 15% CDR.

With Archangle's Staff You can Build more Mana/Abitily Power and adding mor eMana to your self gives you more AP. also When you Spam your Spells you get 4 More Mana each time you Use an Abitily, (3 Second Cooldown) Now With Archangle's Staff you get 3% of your Mana to your AP Soo every I think its 10 Mana( DOn't remember) you get One AP.