You can get downtown pretty quickly from Kastrup (the airport) by train/subway train. Or taxi. If it's your first visit maybe head for Nyhavn, a cute little part of town by a canal where you can grab a beer or a hot waffle with hot chocolate. It's also close to Ströget shopping street and Kongens Nytorv square (which was still a construction site last time I was there). If you want to go for a walk or sit in the park maybe you would just make it to the famous Little Mermaid statue around the corner by the fort. It's about 20 mins walk from Nyhavn. Look for Kastellet (the fort) and Den lille havfruen (the little mermaid).

If you just want to see Copenhagen you can always head up to Sticks 'n Sushi on top of Tivoli hotel and conference center. They have a rooftop bar, great sushi with views over CPH through huge windows, and you can have your dessert and drink in a swing overlooking the city.