If you're talking APS-C sensor size, an 85mm f/1.8 (or the almost identical 100mm version) might be a good option. 50mm could be too loose-framing unless you can walk right up to the horse (and I kinda doubt it, since it's a show), but if you have a little room you should be able to take a full-body portrait. 135mm would give you the ability to produce amazing images, but probably just a head portrait at a distance - on APS-C that is really tight framing. It's also well more expensive than $500 - I was able to get one for about $870. They sell new for $1000+ before shipping.
I would take along a simple, basic zoom lens and note how well it frames. You don't need to take any photos, just imagine (if there aren't any appropriate sized subjects) there is one there and frame the scene with the lens accordingly. If you can't to go to the place the show will be held, try to guess at the distances. If you find that the basic 18-55mm lens covers it well, then you are set - a 50mm f/1.4 will be a good bet (at the wide end), though maybe not perfect on sharpness (especially at the corners wide open - stop down as much as possible). If it's a bit loose, you might move to the 85mm. If the long end of a 18-55mm zoom is way too short, then maybe it's time to think about the 135mm. The 200mm and longer options seem like overkill to me.
I've got a 120-300mm f/2.8 zoom lens but I've discovered that it's no replacement for an 135mm f/2L, even if it's just as sharp. Ease of handling and the bit of extra light are big factors for the 135mm f/2L.