Consider staying in central Copenhagen. I'm not seeing much in the way of hostels right at the airport. And as Edgar points out, the airport isn't that far from the city center, and they're connected by frequent train service.
To make your early-morning departure easier, I'd look for a hostel near either the main train station or a Metro station. (Both the Metro and the regular trains can take you directly to the airport.) You should have plenty to choose from.
As for what to do - what kind of things do you like to do? Tivoli Gardens, a 19th century amusement park, is right across from the main train station and is open all evening. Or you could walk over to Christiania, a "colorful counterculture squatters' colony."