Is a Arabian Horse good for western riding?

It depends. Any breed can be a barrel horse, its just whether the horse loves to run and has the desire to do it. If it doesn't have the desire, it will be a pain trying to force it to run.

They can for a starter horse, but to get to the upper levels, you would need a horse more like a quarter horse, with good hindquarters, built for sprinting and turning on a dime. And the desire to barrel race is important too.you won't do well on a horse who doesn't like his job because he will protest all the way.