If the airport hotel is not taking bookings, I'd just stay in the airport (airside). It doesn't seem worth it to mess around getting to another hotel for 3-4 hours of sleep. At those times I don't think there is any way out of the airport apart from the overpriced coach to city centre, overnight coach to other Swedish cities, or taxis. If it wasn't the middle of winter I would consider taking a coach 2 hours somewhere and 2 hours back if the timetables aligned, or even just walk around the city.