While you likely have access to the very decent SAS lounges at CPH - do not spend the time at the airport. Go to either of the cities.

Thre trip to Malmo is described above. Going to Copenhagen will set you back around 30 kr each way (you need a 3 zone ticket, which is valid on buses, trains and metro). You can either take the train, which will bring you to Copenhagen Central Station in about 15 minuttes (third stop, trains leaving from underground Terminal 3 - same place as the Malmo trains) or you can take the Metro which will bring you to Kongens Nytorv (close to Nyhavn if the wheather is nice) in about 15min (trains leaving from the oposite end of Terminal 3 as from where you exit from customs, but one floor up, look for the M sign). Plenty to do in Copenhagen.

Take a look at http://www.m.dk/en/welcome

Prices in Copenhagen and Malmo is approx identical. If you want to go shopping you are likely to find beter deals in Malmo. For dinning the vote is still out, but for a drink you definately want to stay on the Danish side.