How to Know if a Horse Will be Good in Barrels?
Probably not the answer you were looking for, but it's the truth. Yes, there are things you can look for to increase the chances the horse will excel at barrels (such as the conformation tips above, or bloodlines), but there's no way to predict what's really going to happen.
Some say barrel racing is "the constant search for a faster horse". And some say you can train 10 horses to run barrels and maybe one of them will turn out to be "the one". I believe there's truth to both of those.
Don't worry if the horse decides he/she doesn't like barrels. Or isn't fast enough to win. Or whatever. Because there's really nothing you can do about it, besides buying a finished proven horse (which you stated you can't afford ... most people can't, myself included).
∞•*˚ Βгįťţαňγ ˚*•∞
It is not enough to know how to ride; one must know how to fall.