You can take a taxi from the airport to Stockholm City for a fixed price around 500 SEK. There are also express trains for a cheaper price around 300SEK or the airport buses for 120 SEK. If you want to travel with the municipal transport all the way this will take you a bit more than one hour. You will take bus 583 or 579 to Märsta Station. Then the commuter train to Stockholm Central. After that two stations with the subway to Fridhemsplan. This ticket costs you 44 SEK.
