I take 20 to WAS and points north from CLT pretty regularly, and I have never once still been awake for GRO. On the bright side, they have a pretty nice station in Greensboro where you could probably find an open bench to attempt to sleep for a bit if you were so inclined. There are also several nice restaurants and bars within easy walking distance of the station where you could kill some time/grab a meal after getting off 79. You'd still have a good 3 or 4 hours to spend in the station after those close though.
The star arrives in WAS(if on time) a little less than 1.5h before the departure of the Cap which puts it outside amtrak's normal guaranteed connection window. You can always call the reservation number and ask very nicely if they can book it as a straight connection since it's so close to the 90 minute mark. You might get lucky. You could also live dangerously and book them as two separate segments but then you're hosed if you miss your connection. If you can pay for the flexible fare for the cap segment that might be worth it though.
