For good airport acces, stay near one of the Metro stations, Nørreport st, Kongens Nytorv st. and Christianshavn st. are all conveniently located near things off varoius interest. It's about 20 mins to get to the Airport by the Metro.
I can't comment on individual hotels, since I live here i've never used a hotel in Copenhagen. There are a few budget hotels near Nørreport and the ones near Kongens Nytorv are more upscale.
Here is an example of a walk you could do, if you where staying near Nørreport: You could check out Rosenborg Castle in Kongens Have, Rundetårn (the round tower) on Købmagergade, follow Købmagergade (pedestrian shopping street) to down to Højbro Plads and further on to the Parlament (Christiansborg) and the old Stock Exchange (Børsen), from there walk down to Kongens Nytorv and see nearby Nyhavn. Then take the metro from Kongens Nytorv to the Airport. If you stay near Kongens Nytorv, do it the other way around.
Metro tickets are valid for a time period, not individual journeys, so you could pop back to your hotel, get you luggage and hop back on the metro to the airport on the same ticket.