If you’re staying at the CAB Inn Scandinavia in Copenhagen, and need to get to/from Kastrup airport, there’s a really easy way to do it, a way that avoids the crowded downtown Central Station and any need to walk long distances, or take cabs or buses.
From the airport to the hotel: At the far end of the arrivals terminal buy a “3 zone” transit ticket, either from the machines or from the wickets; at this writing the cost is 25,50 Kroner. Go downstairs and catch the Copenhagen (København) bound train. Don’t go all the way into the city, however: get off at Ørestad. Walk upstairs and get on a Vanløse-bound Metro. You don’t need a new ticket, as your “3 zone” ticket for the train is still in effect. Stay on the Metro through to the Forum stop and get off. Walk upstairs. As you exit the Forum Metro station keep walking straight ahead in the same direction along the main street you emerge on; half a block later, just after the statue of three green men, turn right. CAB Inn Scandinavia is 1/2 block ahead on the right, very obvious. The entire trip, assuming good connections, will take you 30 minutes airport to hotel.
From the hotel to the airport: At the hotel make sure you have 25,50 Kroner in change; if not, ask at reception and they will help. Walk out of the hotel and go to the left. At the main street, in front of the statue of the three green men, turn left. Just ahead you will see the Forum, with the Forum Metro station in front. Walk to the station. Take the stairs down one level and buy a “3 zone” ticket from the machine, at this writing it’s 25,50 Kroner. Walk down to the platform and get on a Vestamager-bound Metro. Ride the Metro to the Ørestad station. Get off and walk to the end of the platform (away from the large Field’s shopping mall) and down to Track 1 of the train platform below. You don’t need a new ticket, as your “3 zone” ticket for the train is still in effect. Catch any train bound for Kastrup airport; there are helpful video displays at the track that show upcoming trains. The train to the airport takes about 8 minutes. Total trip, assuming good connections, will take you about 30 minutes, hotel to airport.
Note: at some point in the future, the Metro will be extended all the way to Kastrup, making this an even simpler transit. Watch the Metro website for updates.