The quickest way is to grab a cab. It takes arround 20-25 minutes to Johanesbergs Castle from Arlanda airport.
From stockholm: Take the E4 and turn of into road nr 273 that leads to Johannesbergs Castle. It takes about 55 minutes.
From Arlanda - Take road 73 and drive for about 25 minutes.
From Gävle: After uppsala, take road nr 77 and follow this for 20 minutes.
From stockholm - Take commuter train 14 to Mörby central. Get of at Danderyds hospital. Take bus 639 to Rimbo station. Change bus to 677 to Johannesbergs Castle (slott).
From Arlanda - Walk to Tomta station on the other side of the E14 (about 16 min walk). Take the train to Knivsta AR terminal then take bus nr 677 for 23 minutes
From Northern Sweden: Take the train to Uppsala station. Then bus 677 to Johannesbergs Castle.