Garrett,
Please try to play Elie while you're in St. Andrews. You will not be sorry. I have not played the Castle so I'm not qualified to say it's a 0, however, you will have a blast playing Elie. I get beat up over this constantly from my Scottish friends but I really like the Craighead course at Crail. It looks like a links course and plays like one if it hasn't rained in a while. Mr. Lennie at Crail is one of the great golf pros in the world of golf.
The month of your trip, you're going to probably need an assist from a member to get on Muirfield. You have a chance if you stay at Greywalls but you should contact them sooner than later. Your problem is that they book up a year or more in advance and you have to have either two or four players if you're unaccompanied. Tuesday and Thursday are visitor days. Morning four ball, afternoon no more than two balls, usually alternate shot.
Gullane is worth playing. However check out Renaissance (I have no idea how hard it is to get on but very nice course), Dunbar or Goswick over Luffness New. It's pretty short and not really memorable. Make arrangements to meet Mr. Baird when you're in Gullane. The pro shop there should have info. I don't know if his little museum is still there.
The only other course I have not played is Fraserburgh. The commentary and pics are a fantastic endorsement for the place. Your first trip will be awesome. Your second trip should include Machrihanish, Prestwick, Turnberry, and Machrie if time permits.
Have a great trip.
Steve