ok, that should be fine even if you miss a stud (no guarantees, just make sure anything beneath it to be durable as well)

so the choice will be whether you want anyone looking at how you hung it to be impressed by your craftsmanship or your idgaf

for my idgaf approach, I'd have a level handy, get one screw in, then make sure it's level before putting in the rest (a pencil line on the wall with the level can give you a good approximate height and a "level enough" but keeping the level on for the second screw is barely any extra work)

From the direction of the holes you showed, I expect the previous owner (and probably the one who made it) screwed drywall screws diagonally in through the sides.

The low effort solution would be to reuse those holes and do the same thing.

No matter what hanging advice you take, please don't put anything heavy/breakable on it because the build quality of the shelf could be just as low effort as the previously mentioned hanging method