It's normal to be nervous at a horse show... It used to happen to me!
Some people listen some music before they get into the arena because it helps them to be focused on warming up or on checking the course correctly. Also you can go to a horse show with a friend or someone of your family who can cheer you up before and after you get into the arena... Being around people you love can relax yourself and maybe you can have a good time with them too!
In my case, the problem was when I got into de show arena and started jumping the course... I didn't breath at all! So I just walk out the arena super dizzy... My trainer recommended me to sing a song during the test, because it will make me breath for sure.
Good luck on your future shows!