Copenhagen is a fairly expensive city for hotels. But there are options in that price range.

My go to hotel, which is often in the higher end of your price range is AC Bella Sky (AC by Marriott) it is 5 to 6 minutes walk from the metro, fairly easily accessible from the airport and a quite decent hotel. There is a chain of hotels called Cabinn hotels, cheap, practical and well located that will get you accommodation at the lower end if your range.

For a bar/restaunt area I would suggest Kødbyen (meat packing district) while not Danish food, it is a very nice place to go for great food and drinks.