Normally I would say, shine comes from the inside out, and suggest feed supplements and such, but since he's a school horse, you really don't have control over that.
Most shine sprays I know of, are usually oil based, so I always make sure I tack up first, and then spray, the few times I use them. I try to stay away from any sprays like that. If I use them I make sure the horse gets a bath not too long afterwards to wash it off, since it's not the best to the coat. But there is Showsheen, Champion Pepi, Ultra, Laser Sheen, Cowboy Magic, just to name a few brands.
Lots of elbow grease, and currying. Use all your brushes (curry, stiff bristle, medium, then soft, etc.), and a soft rag lightly damp to do a final run over to get rid of the remaining dust.
If you can, see if he can get a bath a day or two before the show, to loosen up everything, and get the oils flowing to his coat. You're really probably going to get the best out of just a good grooming.
There are several different kinds of hoof oils. Some, are for conditioning and maintenance, some to soften hooves, some to firm up hooves, and then there is the dressings, that usually come in either black or clear for the shows. You don't have to use it, but if the horse has any hoof issues, or anything like that, the different dressings can help with it. If the horses hooves are fine, then there's no need for it really, unless you just want to give a little support with a conditioner.
I use Hooflex for conditioning, and the black and clear hoof polish depending on the show, also made by Absorbine. Most feed/tack stores should carry several different brands.
"You're just as sane as I am."~Luna Lovegood.