I tried to add website addresses for you but trip advisor did not allow me to do so. Please go to Brewery website, Salzburg Tourist office and Salzburg public transportation sites.
You can take bus Line 1 or 10 from Salzburg AG, bus stop "Bräuhausstraße". I took it from the bus stop just across the bridge. You can also contact them via email - they are very informative and good at communications.
Also Salzburg tourist office is very good at communications.
I think it takes about 15 min. from the bus stop. Be sure to tell the driver that you would like to visit the brewery in German (safer he might understand in English but I spoke only in German).
Enjoy - it is fun.
PS - I also 'liked' their website. You can communicate with them via FB as well.
Cross the bridge and walk to Mirabell Platz (just outside Mirabell Gardens) and take bus no. 1 which will drop you near the brewery on the main road. The brewery is around 10 minutes walk from here. Ask the driver to let you know when the stop comes as it is at a quiet place and not touristy at all. Pls note that the hop on hop off buses DO NOT go this brewery.
We took a taxi from the brewery to the Fortress for about 10 euros. The gift shop call the cab for us.
Get quick answers from Stiegl-Brauwelt staff and past visitors.