Not really, but few experiences could be worse than hanging around in that airport overnight.

There's not much to do in Doha at any hour, especially at that hour. I would recommend taking a taxi to Souq Waqif (which will cost less than USD 10) and sitting at one of the Arabic restaurants/cafes. None of them have great food or service -- and a couple are open 24 hours -- but it's a nice environment, and at midnight when you arrive, there will still be some activity. Then take a taxi back to the airport when you're ready, though taxis can be hard to find so allow plenty of time.

The bars in Doha are open until 2:00. There are decent bar/lounge options at a few hotels near the airport, especially the Sharq Village or Oryx Rotana.