'Flygbussarna' used to have a direct service between BMA and ARN, but as far as I know that one is history. They still operate BMA to 'city terminalen' where one can change for another coach service or the express train. The coach service would cost 69+89 SEK and 69+180 SEK for a one way coach+train trip.
Taking a cab would probably be the fastest way.
BTW, 'Arlanda Express' has now opend their new escalator between the train station and terminal 5 north