Yes, if your flight is on time, you should have plenty of time to see a bit of Copenhagen. Copenhagen airport is not too big (by international airport standards, anyway), and it's not too far from the city center, so it's pretty easy to get in and out.
The airport is served by both the Metro and the train system. Which one you should take depends on where you want to go, which depends on what you want to do.
You can take the Metro to Kongens Nytorv, which is right next to the famous harbor Nyhavn. (If you've seen pictures of Copenhagen with the rows of brightly colored houses along the water, that's Nyhavn.) From there, you can take a boat tour of the harbor, which will give you a glimpse of the Little Mermaid statue and some other famous sites. Look for the "Netto" company instead of its more expensive competitor. Their tours cost 40 kr (about US$6 - a bargain), last about an hour, and leave frequently throughout the day.
Or you can take the train to the main train station, which is 200 m from the Tourist Information center. There are walking tours that leave from the TI at certain hours of the morning, which may be of interest to you depending on the timing of your flights. The train station is also across the street from the Tivoli Gardens amusement park.
Alternatively, Kongens Nytorv, the main train station, and Norreport station (served by both the trains and the Metro) are all good starting points to walk around the city on your own, if you have a good map and know where you're going.