Will you already have a car, or would you be renting it for a day in Salzburg just to make the trip and then turning it in in a different country when you get to Passau? The trip by car to Passau takes ~2½ hour according to ViaMichelin, but renting it and returning it will take time, so the whole trip will take well over 3 hours.
By rail you can go from Salzburg to Wels, Austria, on Westbahn for 10,40€/P, then from Wels to Passau for 18€/P. Total time would be 2½ hours, as fast as driving without having to take the time to rent and return the car.
Slower, but cheaper, would be to go by regional trains via Mühldorf (through Germany) using a Südostbayern-Ticket (regional pass) for 17€ single, 22€ for two. That trip would take 3h41m; the extra time might be worth the savings (17€ vs 28€ single, 22€ vs 57€ for two).
In each case you would need an extra leg to get to Vilshofen. Vilshofen is very close to Passau and a pleasant little town. I stayed there in 2007, on my way to Burghausen via Passau because I found Passau to be a little pricey vs Vilshofen.