I've spent on a few days in Copenhagen the last 4-5 summers after the Roskilde Festival. Always on a budget, so I should've had several nice tips for you - I only have a few random ones.
- For a cheap take-away lunch, Vandkunsten Sandwich
is probably not worth going out of your way for, but it's perfect if you are in the area. It might have been the circumstances, but it's the one of the few memorable sandwiches I've had. Really tasty and of very good value.
- Jeppes Badehotel
brings fuzzy warm memories. It's just a small, cozy bar with a funny old bartender.
If you have a day off Louisiana Museum of Modern Art
is well worth the trip. It's about 30 minutes by train, and another 15 by foot. The building and its surroundings is just as worthy of a visit as the actual art exhibition.
Other than that there are lots of cheap buffets, but none have made an impression on me. I haven't had any memorable shopping experiences either, I'm afraid.