Go to the Qatar E government site, get a transit visa and definately leave the airport. There will be zero open that early and your only time killer will be a walk on the corniche along the water or a visit to 'City Centre' the city's largest and most modern shopping complex. I'd wait in the lounge for an hour or so then leave. The lounges as with the airport will be terribly boring but go through the duty free area and shower n rest in the lounge before u leave. Good luck.