If I were you, I'd fly into Aberdeen, rent a car and drive to Cruden Bay. Get a room at the Kilmarnock Arms. It's across the road from the third green and fourth tee. It is also famous for being were Bram Stroker wrote
Dracula with inspiration from nearby Slain Castle. It isn't a luxury hotel, probably cost around £30 or £40. The floors aren't level and it's shared facilities. But the golfers all hang out in the bar at night to talk about their games. Buy a drink or two and your bound to setup a game for the next day. When I was there last the son of the owners of the hotel, James, was the club champion.
You'll want to play Cruden Bay at least two or three times. When I was there it was an £8 rate for the entire day if you played with a member for that first round.
I think it is probably a little longer than a 2 hour drive to St. Andrews, but it's a nice drive. Aberdeen is a cool city for site seeing.
However, before doing all this you might want to check with Cruden Bay, make sure they aren't having a competition that day, or if they are, if it is open. Also if you don't have a match, let the pro know you'd like to play with locals. I'm guessing much of Saturday play is now tourists.
Re-reading your post, you might be better off at Cruden Bay on the Wednesday or Thursday on your way out of Scotland. I made the drive once from Cruden to Glasgow and don't remember it being that tough of a drive.
If you go to Cruden at the beginning of your trip you might want to stick around one more night and get an early round in Sunday morning. No point of rushing to St. Andrews, since TOC is closed.
Dan King
There are burns to cross, hills to carry, and hidden nooks to drop into. Some of the shots are blind -- willfully blind if you like to call it so...but there are some truly fine golfing holes, on the grand scale.
--Bernard Darwin