Hello!
I will have an 18hr layover in Copenhagen in August. I arrive at 1pm. I'd like accommodations, hostel, close to the airport. My flight is at 7am the next morning. Also, I should have a good 4-5hrs to check out the town. Any recommendations about what to see or do with this limited window of time?
Train from the CPH train station to Copenhagen's central station runs about every 10 minutes and takes less than 15 minutes. Cost less than 40 DKK.
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."
If you want a basic hotel very close (1/2 block) from the central train station and low priced for Copenhagen, try the Hotel Nebo. Standard doubles with ensuite at over $100 USD with economy doubles with down the hall bath facilities under $100 USD per night per room. Economy restaurants short walk from the Nebo. Local buses seem to hub at the train station for some reason.
http://www.nebo.dk/
Note that the train runs less frequently in the very early morning (20 minute intervals?). The train departing at 4:00 gets to the airport at 4:12. Transportation planner:
http://www.rejseplanen.dk/
Enter Hotel Nebo as a starting point, Airport as destination and desired arrival time. Look for the train option (planner also shows bus options) and it will include walk time from the hotel.
If you want a hostel check:
http://www.visitcopenhagen.com/copenhagen/transportation/hostels
The Backpacker Hostel is near the central train station and had rental bikes.
