score:3
According to resrobot, you can take bus 152 to Solna, then a metro to Kista from there.
Details will depend on exactly when you're travelling. You may have longer waits or have to take several buses when travelling in the evening or weekends. Just type the exact time, date into the resrobot travel planner (or the local transport company website SL.se, as Crazydre reports it may be more up-to-date) to search for connections from Bromma to Kista.
Upvote:0
To get from Bromma to central Kista: Take bus 152 to Solna centrum, and change for the blue tube line (direction Akalla) to Kista. If going to eastern Kista (including the exhibition centre), the 178 bus runs frequently from the Kista tube station to that area.
To get from Bromma to eastern Kista: Take bus 152 to Karlsbodavägen or 110 to Lintavägen. Then take the tram to Solna station. Finally, catch a suburban train (direction Märsta or Uppsala), and get off at Helenelund (two stops), where you can catch bus 178 to (for example) the Kista exhibition centre.
To get from Bromma to Södermalm: Take bus 152 to Sundbybergs torg, then suburban train (direction Västerhaninge or Nynäshamn) to Stockholms södra (located near Mariatorget and Medborgarplatsen)
To get from Södermalm to Kista: Take the suburban train (direction Märsta or Uppsala) from Stockholms Södra to Helenelund. Bus 178 runs frequently between Helenelund and Kista, which is only a short ride.
To get from Kista to Södermalm: Take bus 178 to Helenelund, then suburban train (direction Södertälje, Tumba or Älvsjö) to Stockholms södra.
Upvote:2
Just adding a couple of cents since I have done similar trips to this particular airport a couple of times.
At the moment 152 is the only bus that accesses the airport directly from this general direction. (There is also 110 from Alvik - thanks Crazydre for pointing it out.) But I think it may be good to point out the tram alternative. (That you also may get from resrobot searches.) From the tram stop Karlsbodavägen it is a bit more than a kilometre to walk. Depending on where you are in Kista it may make sense to take the commuter train from Helenelund to Solna instead of the subway.
This will improve vastly in the future. There is a new branch of the tram planned that will pass the airport and then head for Kista. It is planned to be completed 2023 so you'll have to wait a bit for that....