Does anyone know if many shops or attractions are closed on Sunday in Salzburg?
Attractions, like museums and restaurants, are open. Stores are usually closed. This is in all of Austria and Germany, not just Salzburg. If you need to buy something on a Sunday, head to a train station or airport. The stores there are allowed to open longer hours and Sundays, and of course tend to be more expensive because of this.
Sunday is still considered the day of rest. I believe the law is if something can remotely be connected to traveling, it can be sold. In addition to the train station, don't forget "gas stations." Some offer an extensive line of drinks, snacks, sandwiches, etc. in addition to maps, magazines, post cards, etc.