Congrats! Though I have not done it, Greenbrier regulars say take a half day and drive in to Lewisburg. Good restaurants and shops I'm told. You can search around Tripadvisor for specific recommendations.
We did the carriage ride last time and thought it was fine but expensive--though you do get a nice "History of the Greenbrier" coffee table book as part of it. I am not sure what outdoor activities are going now given it is December. The Greenbrier website posts their daily newsletter so check it out to see whats been going on.
You may have some down time considering you say you're not going to gamble/eat the whole time. My advice is walk around as much of the place as possible. There is so much history all over. If you like golf make your way over to the golf clubhouse where Snead's is. It could double as a muesuem of golf history. It hopefully won't be too cold to walk the grounds and see the landmarks, and head up to the art stores on top of the hill if there're open. Shopping is ok in the main resort but nothing too special. Enjoy!