I was recently found in Copenhagen for about 6 hours waiting for my next flight so I had already decided to take a walk downtown and not wait at the airport. It was such a wise decision! All necessary information can be found here http://intl.m.dk/#!/about+the+metro/travel+information/airport
I had already booked a day pass for 75 DKK (almost 10 euros) to/from the airport by providing my mobile number. So as soon as I landed, some 5' later I got the message giving me a pass for a day. The only metro line to be used is M2 so no need to worry about which line to take. In 13' I stopped at the station Kongens Nytorv and after 1' I was already walking across the wonderful Nyhavn! So easy as such! I had a huge walk around the city and was in time back to the airport to continue my flight!