The BT definitely operates at least some routes after midnight, my roommate drove for them the last 2 years I was there
Only on Friday and Saturday nights. "Late Night Service" goes on until 2am (and buses after 10pm are party-town, that's discussion for another day!)
Sunday night service wraps up on all routes around 10.30pm. Check out BT website for exact times. I did not have a car all through my days at VT and knew schedules of every single route, just confirmed online, not much has changed. If you arrive by AmBus at 2340 on Sunday night, you're out of luck as far as BT connection is concerned.
When I was there we also had weekend busses that ran from Blacksburg to Charlottesville to the Vienna metro station on Friday and Sunday. Straight up 81, 64 and 29 - quick and cheap, I took advantage of them regularly (until I finally got a car sophomore year).
Convenient yes, cheap no. As I said, I did not have a car so I have tried every means possible to get out from Blacksburg. The weekend bus to DC was $54 one way, $102 round-trip in 2010-2011 period. Greyhound from Roanoke to DC at that time used to be $39-44 depending on when you book. Then Megabus came in (just as I was graduating) with $1 fares and even now you can get $15-20 fares on any day. I checked Amtrak for a dummy booking, Blacksburg-DC is being offered for $36. Given a choice, I wouldn't mind waking up at 4.00am for this