Catch riding is when you are hired at a show to take a horse in a class even if you haven't riden it before.
I think to catch ride you will need to be well known at the shows you are going to. You will also have to have already proven your ability. Catch riders show in what ever class they are asked to. If you decide to do it you had better make sure you can get the horse through the class well or you will end up with some angry owners and you will not likely get more jobs.
Ditto to what LauraB said! You will need to be very very experienced by the time someone wants to pay you (or pay for you) to show their horse. They're wanting you to show their horse so they can win, if your abilities aren't to where you can bring out the best of the horse's potential then there's a good chance you'll need some more experience under your belt. There is no age limit to catch riding. I know a lot of kids who are asked to show people's ponies. In fact, I've had a lot of kids show my ponies since I'm too old (and unable to show in amateur classes). If catch riding is what you want to do, I'd try ride/exercise/show as many horses as you can to get your name out there. It's a nice way to get to show horses without having to pay for it, but unfortunately getting to the point where you can do it is usually not cheap. I would suggest talking to your trainer to see what you can do.
I catch ride at my local shows, and you need to make sure you are COMPLETELY ready. Its a difficult thing to do, as you may only have a couple minutes and one or two jumps with the horse before you show. Some horses may be great, others could act like complete brats. And if the class doesn't go well, the owner may blame it on you.
You want to ride as many horses as you can to prepare yourself, and to get your name out there try putting an ad up in your local tack shop saying you can school or exercise horses for people. If you do a good job doing it for free, you may get asked to show the horse, or they may offer to pay you. I schooled and showed horses for free for 3 years before someone offered to pay me. And even now I still school horses for free most of the time, and an owner may pay me $20 to show their horse over 2 jump courses and a flat class. It takes time to get up there, but if you eventually want to be a trainer, then its great experience.