Three hours in Salzburg

You can go up the funicular, see the view, and come back down in 30 minutes easily.
I'm not sure how long Rick's old town walk is supposed to take but, if you have 2 1/2 hours, you can easily walk the old town. It isn't large.

we did the rick steves tour on our phone/ipod, it was good.( as usual) the highlight for us was ( while not that far from the funicular etc) was to walk around to the abbey that was filmed in the sound of music. it was exactly how it was filmed in the movie. and just tucked in an area that didn't seem like tourists know it was there ( at least when we were there)
the beautiful little alleys/shops in salzburg is somewhere you could spend a lot of time in. we unfortunately only had one day. would love to go back.

I didn't follow Mr. Steve's walk, but it didn't take me long to thoroughly explore Salzburg's Altstadt. As noted, for all the tourists it attracts, the old center is surprisingly small, not much larger than a decent-sized shopping mall (the more modern city, however, is much larger).

Why do you have three hours in Salzburg? Between trains or between flights?

If between flights, no time. If between trains, yes, but pay attention to the time. The buses from in front of the Hauptbahnhof go all directions, but most, if not all, go first to the old town.

We are doing a Viator tour so we have three hours on our own to see the town.

Yes, you will have time if you move through fairly quickly. I would plan an hour for the funicular/castle and two hours for the Old Town area. ...just enough time to want to return again!