Barkarby is free of charge, if you stay there just a couple of nights. Beware that occasionally the runway is closed and used for drag racing or other motor sports. Not sure if this is published in Notams.
The local flying club have Avgas 91UL and 100LL for sale.
There is a home page in Swedish with some maps of the airfield.
http://www.barkarbyflygklubb.se/eskbregler.asp
But on the other hand you need to order a taxi from Barkarby to wherever you want to go. Quite far and expensive. From Bromma you can take the bus for 20 SEK (app 2,20 Euro). I can check how much the landing and parking fee at Bromma would be on Monday...maybe it would be a better option anyway.