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
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.