In early June we are arriving in Copenhagen from a cruise on a Saturday morning. Our flight is at 7am the next morning. What kinds of transportation is available that early in the morning to the airport from the city center? We would rather stay there rather than near the airport to do some sightseeing on Saturday.
There are very easy connections from Copenhagen Central Station to the Airport. Check out http://www.rejseplanen.dk/bin/query.exe/en which now has a schedule up through April (but gives you an idea how frequently the city metro goes.) I think during the day, the train between the city and the airport goes about every 20 minutes. SAS used to have a bus from the City center to the airport; it's worth checking to see if this still exists if you are flying SAS. I suggest staying in the neighborhood of the Central Station because you are Tivoli-adjacent, a couple of blocks from the walking street Stroget and not far from museums, the Botanical Garden and Rosenborg Castle. I can't think of a better day in Copenhagen than any of those followed by an evening at Tivoli! We always stay at Hotel Absalon, which has both a budget and a more expensive wing. In the budget wing, the bathroom is down the hall but the rooms are fine and breakfast is included (maybe not if you are leaving so early, though.) I envy you your trip-- enjoy!
I am headed there this coming fall --- The M2 metro will take you from Kongens Nytorv to the terminal in under ten minutes .