Having been to DOH before, I think what I'd do is book in at the airport hotel to tide you over until lunchtime; grab lunch at the lounge, get a taxi to the massive Venice-themed shopping mall, head back and see the museum ~3pm, while away an hour or two at the Souq, and then get a sneaky dinner in at a nice restaurant. I wouldn't want to be wandering around Doha on 2 hours sleep.