If you're looking for a more budget option in Copenhagen, I like the Wakeup hotels - there's one at the back of the main station, and one on Borgergade, a short walk from Kongens Nytorv. Both can be easily reached from the airport: the former by train; the latter by metro. They're well-designed budget hotels - fresh, tiny, but well designed rooms.

(The area around Halmtorvet, near the station, does have a number of decent and reasonable-for-Copenhagen eating options, particularly if you're dining solo.)

(FWIW, the airport is pretty efficient and the city small, so, knowing it well, I have no problem leaving my office in central Copenhagen 90 minutes before a short-haul flight is due to depart...)

In Malmo, I like the Moment Hotel, opposite the station. Again, tiny rooms, but a nice, fresh refurbished old hotel.