4 hour layover - whàt can i do?

The metro will take you from the airport to Copenhagen i 20 minutes, so a short trip should be possible. Take the metro to either kgs. Nytorv and have a look at Nyhavn (colorfull houses along a canal with lots of bars and restaurants) or to Nørreport for stroll around the city centre and a look at Kgs Have (kings garden) and Rosenborg Palace.

But figure 20 minutes into the city, 20 minutes back, and your four hours is now 3 hours and 20 minutes. I would arrive back at the airport no later than two hours before your flight, so now you are down to 1 hour and 20 minutes. And that is assuming you deplane and get through customs immediately (unless you are coming from another European country, then you wouldn't have to go through customs). IMO, 4 hours is not enough time to leave the airport.

Dragor is a small fishing village near the airport. It has some shops etc to visit and view of the bridge to Sweden. You can get there in about 15 minutes. Catch Bus 35-I think just out the front door of the airport. Bus runs about every 30 minutes. You pay bus driver for ticket. Not Copenhagen but easier to visit with your limited time but may still be cutting it close.

I'd say 20 minutes is a conservative estimate from the Airport to Kongens Nytorv, I'd have guessed 15. The metro gets you there faster than the train gets you to the center.

I'd plan two things, Rosenborg palace - both upstairs and the crown jewels in the basement. It's a few blocks walk from Kongens Nytorv, closer to the Norreport stop. Afterward pick up picnic stuff at a grocery store near there and picnic right around Nyhavn (which is directly adjacent to Kongens Nytorv). You can adjust your picnic time based on when you need to get back. If you have extra time, stroll part way down the Stroget (the north end is Kongens Nytorv) before catching a metro back to the airport.

There are lots of other sights but I think this is worthwhile without being rushed.

Not to scare you, but on my last trip to Copenhagen we had a morning flight. Took the train from the airport and had our plan to do the same when we departed a few days later. Arrived in plenty of time only to find the train was not running due to some electrical problem???? So, we had to quickly try to find a taxi along with all the rest of the travelers who were stranded. Luckily we made it in time to catch our flight to London....
I would not risk it with such a little bit of time.

The Denmark Aquarium is the closest major attraction to the airport. You could reasonably expect to spend a couple of hours there. With such a short layover, you might consider eating at one of the excellent restaurants in the airport- aamans has a cafe there that is excellent. While the public transportation is excellent, a four hour layover is very short, especially if you are flying internationally.

While I have visited Copenhagen at a more leisurely pace, I have done something exactly like this recently. I took the train in from Kastrup and was able to get to the Little Mermaid and back to Tivoli for a quick beer. Tivoli is very close to Central Station. While it was quite and adventure, I knew the entire time I was missing out on a lot which was confirmed later. However, that brief taste of Copenhagen definitely made me want to come back. Give it a shot.

My wife and I were in Scandinavia and the EU for a 35 day visit last year. There were lots of "moving parts" to this journey and if we learned nothing else from this adventure, it was "don't cut schedules too close". One such occasion (of many) was a flight from Stockholm to Oslo, with a connecting flight to Kirkenes. We had 2 hours between flights, as it turned out our flight was delayed in landing (for no declared reason) and our luggage was late in making it off the flight from Stockholm. We ended up running the entire length of the concourse to make the plane to Kirkenes--our luggage didn't make it and almost didn't arrive in time for us to board a cruise ship the following day. On other occasions when we arrived in several cities which had “quick” train/tram connections from the airport to the city, we "just missed" the train/tram, or our credit card wouldn’t work in the automated ticket kiosk, etc. etc. etc. Those 10-15 minute glitches add up quickly.

Load a good book on your e-reader for the four hours or people watch in the airport. With the little time you'll have in Copenhagen, you would have to get a map and mark where you took pictures just to remember what you saw and where you saw it/them. It's unlikely you would have time to "experience" anything. During our visit there last year we spent two hours having lunch outdoors under an awning while it poured rain. During lunch, one of the awnings gave way drenching a couple two tables away--that was an experience; we can recall the restaurant, the server, the way the wind and temperature changed, and the entire balance of the afternoon all with vivid nuances.