In 2002, we came over to Salzburg from Munich for a day trip. We arrived just before noon and walked down Ranierstrasse to the old town, passing and spending time at Mirabell on the way. In the old town, we walked around town, saw the Residenz (where they hung the large Nazi flag after the Anschluß), had lunch, then went up to the castle on the incline. We spent quite a lot of time in the castle, took the tour, then came down the front, over the drawbridge. At the entrance kiosk for the castle, we turned hard right and followed the path, Festungsgasse, around the hill to Nonnberg Abbey. You cannot go into the cloistered Abbey, but you can visit the chapel. (I realize immediately that the chapel is not what was shown in the movie. That was a Hollywood set. Or was it at another church in town?) After that we walked back to the Hbf and returned to Munich in time for a late dinner.
So yes, I think time will permit you to do the fortress and Mirabell. No, you don't need a transport pass (unless as someone else mentioned, you need it to get from your hotel into town).
By the way, before taking an SOM tour, find the DVD of SOM, the one with commentary by producer/director Robt Wise. After listening to his commentary, you'll realize that, except maybe the gazebo, which was only used for distance shots (the singing and dancing scenes were also in a Hollywood soundstage) and is now at a different location, nothing on the tours, except what you can see by yourself walking around, is really from the movie.
Hallstatt is a bit of a trip from Salzburg, but not far (1-1½ hr) off the path from Vienna to Salzburg. Why backtrack? I'd stop and spend a night in Hallstatt on the way to Salzburg. My suggestion:
Day 4 - Narschmark before going to Hallstatt by train
Day 5 - Hallstatt, go to Salzburg.
Day 6 - Hohensalzburg Fortress, Mirabell Palace and Garden