I've spent some time in Freilassing, usually coming from Munich by train and going to or coming from Salzburg by train. I assume you are talking about picking up the car at Europcar on Lugwig-Zeller-Str (right where the street out of Salzburg, Salzburger Str., turns into Münchner Str. It's about a half mile walk from the Freilassing Bahnhof to Europcar, and there are regular trains and S-Bahn from Salzburg Hbf to Freilassing Hbf. If you were coming from the oldtown Salzburg, I would suggest you come out on bus 24; it stops right there at the intersection of Salzburgerstr and Zellerstr.
However, from where I see the Salzburg City HI, it's kind of a toss-up. Do you walk ¾ of a mile to the Salzburg Hbf, take the train into the Freilassing Hbf, then walk ½ mile to Europcar, or do you walk 1 mile to Ferdinand-Hanuschplatz (across the river in the old town) then take bus 24 directly to Europcar.
It look like there is also a bus-bus connection taking O-bus 2 from near the HI to Esshaverstr where you catch bus 24 to Europcar. Less walking, but it requires a transfer.
I know nothing about taking taxis in that place. Maybe take a taxi to the Salzburg Hbf, then a train or S-Bahn to the Freilassing Hbf. Stash your luggage in a lacker, walk to the Europcar place, drive back to the Freilassing Hbf and get you luggage.