First, and foremost, Greene County has a mighty fine public course at Greene Country Community Park, which is about 5 miles east on US 33, or about 10-15 minutes away. The other nearby courses well worth mentioning are:
Walnut Creek: On the other side of Charlottesville, this is quite simply one of best courses in Virginia.
Meadow Greek Gardens: Nine hole course in Charlotesville, on the near side. Good for beginners
Westover Park: 18 hole course in Harrisonburg