Late model Corollas have paint with a layer of clear coat on top.
You might not like what Im going to say but dark colors are difficult to maintain, they require regular washing, polishing and waxing, that being said I do own a black car, yes most car nuts do, and its a passion of mine to keep it looking its best. Dont get a black car if you dont enjoy detailing your car.
What I would do is hand polish it to regain as much shine as possible, you could use a rotary or orbital buffer to save time and effort but theres the risk of damaging the paint if used incorrectly. Take your time and do it by hand.
1st wash the car, then clay the surface to remove all contaminants, apply polish ( I use Meguiars products ), then a layer of wax.
It should come up looking near new, even if the clear is worn down, there should be enough paint thickness to polish up, just make sure you wax often after the initial polish to protect from bird poo and anything else.
Best thing to do if your car gets pooed on, get it off ASAP!!!!