So far I think I am getting used to Lux in Dominion. I go for Boots of Mobility + Cloth Armor > Morello's Evil Tome > Rabadon's Death Cap if I manage to get enough. I use Ghost + Clarity so I can get in position to top lane and prepare for the big assault. After that its a need to have at least 3 CPs secured. Late game if my team is winning I steal caps to move the game faster.

A big problem though is that I have trouble dealing with Akali. She'll eat me alive even with an oracle elixer, and even more so she'll probably target me because of it. What can I do in a 1 vs 1 situation?

You can't do anything against Akali 1v1. Accept the inevitable that she will kill you every single time unless you manage to burst her from full to 0 in one combo....which is highly unlikely.

As Lux Akali will kill you 1v1 every time unless you are a substantially better player.

Your only defense is not getting that close to her. Your Q, E and R all outrange Akali (1000, 1100/1450 for max detonation range, 3000 respectively vs. Akali's R of 800) so just keep your distance or hug the tower, depending on your position. If you land Light Binding, feel free to combo as much as you can but get the **** outta there within the 2 seconds (or ~1.2 if she has Mercs). If she jumps on you, Q her and try to do as much damage on her as possible while putting enough distance to force her to R again. You're gonna die but at least you can try to put some hurt on her and give your teammates time to come help.

Other than that, MR helps against Akali combined with your Shield but I wouldn't waste too much resources on it; it takes a lot of resources to be able to tank Akali and your damage will suffer a ton if you devote too much in it. Maybe one MR item (perhaps Odyn's/Mercs) or some MR/AP items (Lichbane/Abyssal) but chances are you're best off just trying to kite and not running blindly in the jungle; it's generally always a bad idea with long range squishies.

Don't use clarity, use ghost/exhaust.

Try not to get near her Dash range.

I think death cap is overrated in dominion, go for hourglass instead,especially on a squishy. It helps you immensely in a tower dive situation.

But the best you can do vs. her in a 1v1 fight is to land a Q as she dashes, land whatever damage you can while running away. When she dashes again, exhaust her, and try to land whatever damage you can. Pop Hourglass and hope you have some cool downs come up to help finish her off. Something like DFG will help you get more damage on her. You'll probably still lose, but she'll be very hurt. Also remember to use your shield.

My dominion lux build is morello > dfg > hourglass

you don't 1 vs 1 akali as a lux
too risky and she has 3 dashes to spare, plus if she has rylai, gunblade's active or merc's, you're pretty much screwed

if you still persists on 1 vs 1, I'd suggest some mr like odyn's veil or abyssal scepter.

get a buddy with some cc + hextech sweeper or lightbinger to counter her if she's building spellvamp and spellvamp. otherwise, under the mr buff of her shroud, she can just heal back the damage you dished out.

Sorry, akali/jax/talon/poppy are the anti-fun on dominion. I stopped playing the heroes I want to play in favor of tank soraka(who is the only hero I like to play who can deal with said champs) just to **** on akali/talon/poppy every time I see them. Funny enough, I see them every time.

you actually can burst her if you have Lich Bane and Deathcap.

land the Q, them E, Ult, auto attack, ignite.

From what I played Lux in Dominion. Your main focus is long distance harass and pushing waves. I find weird your choice for Boots of Mobility, better let the cap stealing to champions with lots of escape methods.
Onto your question, pretty much what everyone else said. Your best bet is to snare that Akali once she dashes into you and run for your life to your nearest teammate that can assist you. You could maybe kill her with a full combo if she has no MR at all.

I have to agree with this. Lux's kit isn't designed for backdooring well and she's much more suited for creep pushing and defending.

Also OP I do the same thing as Diman suggests and wait AFTER she dashes to you, before you Q your snare out. The reason is early on you find that using your Q too early will result in her immediately flashing into your face and you are caught without any escape.

Let her get next to you, Q, drop a E if you think distance is good and run like hell.

Also, even though you didn't ask for it. I think you should TRY this build. It may not be your playstyle but I've experimented with a ton of builds and find this works the best.

Runes- AP pen quints/marks, blue -CD reduction, yellows -mana regen, health or dodge it really won't matter a whole lot
Masteries - Standard Caster build 10/0/21
Items - Boots of lucidity (15% cd redux), Kindle Gem (10% cd redux), Haunting guise (20 ap pen), Rabadons cap, and whatever the hell you want after as long as it has AP EXCEPT rod of ages.

You'll find that CD reduction is priority when playing her. You become mana starved a lot but pick up health kits. Then go straight for the haunting guise as the AP pen helps so much for a cheap price. Then AP.

I strongly am against Rod of ages. It used to be in every build but I found that the cost was too expensive and she was still squishy. It doesn't justify the cost. Also I've tried morellos tomb and I find it's too pricy and you pick up useless dominion stats such as regen. While on paper it seems she needs it, you should be either teleporting to base, dying, or picking up med kits so any additional mana regen is inconsequential compared to other stats.

And obviously be willing to adjust/adapt your item build if it's necessary