Give your Sim the bookworm trait and read the 3 skill books in the library. But the horse needs to skill up too if you're looking to win competitions. One way I've cheated on this skill acquisition is to make Needs static (at the letterbox) and hop on the horse and just go for it. Horse and rider gain skill at the same time. Your Sim needs to be in a very high relationship with the horse or he/she will get thrown off. Also, in order to complete the jumping skill challenges for the horse, your horse needs to jump over the more advanced jumps, not the beginner jumps.
But, if you're into serious cheating, there is a game mod on 'Mod the Sims', Super Speedy Skill Books or something like that, which allows your Sim to read the books to acquire the skill in a couple of Sim hours.
