I played on my golf team all four years when I was in high school (just graduated last year). Most of the spots on our team were already filled by players who were on the team previous years, and our coach supported the idea of seniority playing a role in who plays. I was fortunate enough to play starting freshman year because our team wasn't very talented.
Outside of swing fixes (which can't be accomplished in a week), here are a few tips that may help your chances of making the team:
If you're trying out at a golf course - try to play the course beforehand so you have an advantage over those who have never played there.
You're registered on a golf forum, so I'm going to assume that you're pretty serious about golf. If so, let your coach know you care about the game. You'll have a much better chance at making the team if you let your coach know that you practice a lot, take lessons (if you do), etc. The more serious of a golfer you sound, the more you'll look appealing to make the team
Dress properly. It sounds simple, but if you dress like a hacker out there it doesn't help your case to make the team.
Try to relax, one poor shot in front of your coach doesn't mean your chances are gone. Everybody plays poorly at times, if you happen to have an off day don't worry about it, just come back strong the next day.
If you can get a hand on Bob Rotella's book - Golf is not a Game of Perfect, read it.
Most importantly, and it's a total cliche, but have fun!