I was there earlier this year and stayed at Hotel Baltzar. There is very little to do in Malmo except enjoy the restaurants and bars around the Lilla Torg area. Just slightly away from the square the food and drink is much cheaper. We made the long walk to the Turning Torso building, there is some interesting architecture close bye but I am not sure if it was really worth the long walk, maybe on a sunny day. You can get a bus from Malmo airport to the train station. Our hotel was about 7 minutes from the train station. You use the same station to get the train to Copenhagen.
In Copenhagen its another long walk to see the Little Mermaid but the walk is worth it along the seafront.