Admont Abbey

Normally I vote for public transport, but in this particular case, considering the short time frame, renting a car would be better.
The only route without backtracking is: Salzburg, Hallstatt, Admont, Vienna. But this is not doable in one day.

Sending sympathy. I visited from Hallstatt for part of the day, but I had a car and drove.

I visited Admont Abbey in 2022. I had looked into tours (too much transit time) and public transportation (too complicated and lengthy). We ended up renting a car and spending two nights in Admont. That allowed us to have a leisurely scenic drive from Salzburg, stopping at Hallstatt along the way.

There is a direct train route from Vienna West Bahnhof to Admont, but not every day. The same is true from Melk. Use the excellent OBB app to sort it out. I am currently in Admont for six weeks, which is not too long if you love hiking and/or mountain biking! You can reach some of the trailheads by public bus, others by a call taxi--but keep in mind that a lot of the hikes are actually scrambles and very difficult. Very little in the "easy" category, or even "intermediate". Many challenging routes for climbers. Absolutely gorgeous limestone peaks. Gesause National Park has an office right in town, about five minutes from the abbey, with all the hiking maps etc that you'll need. The abbey itself has a very good exhibit on medieval religious art, the library is beautiful (and unlike in many monastic libraries, you can stay as long as you like without a tour guide), and the grounds are pretty. Most tourists, especially English-speaking, just stop at the abbey, have lunch maybe, and leave. That is perfectly doable. In the past month we have not encountered any other English-speaking people here, either in town or in the surrounding mountains. Do not come here if you want much tourist infrastructure, or nightlife, though the town is very liveable and there are guesthouses. Do not come here if you need services to be provided in English (and your standard German might leave you confused sometimes, because the Ennsthaler dialect is quite different). Keep in mind that almost everything is closed on Sunday and many things are closed on Monday and/or Tuesday, as well as mid-day (eg the book shop takes a three-hour lunch) or by noon (eg the bakery). Even the bio grocery takes a long lunch break. Make sure the abbey will be open on the day you want to visit, and check the hours. (Right now in May, they are only open 10:30-3:30 during five days/week, although maybe tour groups have other access). You really do not need a guided tour of the abbey, as most everything is well-curated in German and English.