I had a 5-hour layover in Copenhagen on my way home from Helsinki and was advised that, rather than kicking my heels in the airport, I should take the speedy (and cheap) train into the city centre. We landed at 11.20 and I was in the city before noon, so you can take that as confirmation of the ease and speediness of the trains.
The central train station is within easy walking distance of many of Copenhagen's sights, so I amused myself by walking around, more or less at random. I didn't buy anything (apart from a sandwich) because I had no idea what the exchange rate was ... (turns out this was a wise decision: Copenhagen is expensive) but I did take lots of photos. Here's a selection. My top hint for those of you who may have 3 hours to spend in Copenhagen in the future is to go to the new Royal Library, locally known as the black diamond. There are free deckchairs on the water-side of the building (see below) and really beautiful secluded gardens behind the original library on the other side. I'd also recommend seeking out the cathedral which is very pared back but no less beautiful for all that.