The metro runs directly from terminal 3 to Kongens Nytorv. Get off at Kongens Nytorv and walk to Phoenix Copenhagen in around 5 minutes.
If you take a taxi from the airport, it will cost around DKK 250. All taxis run on a meter and accept dankort and most major international credit cards.
Take bus no. 26 heading for Ålholm Plads. Get off at Dr. Tværgade/St. Kongensgade and walk to the hotel in around 2 minutes.