You should be aware that May 29, the day you arrive in Vienna, is a public holiday (Ascension). This means that public transportation will run on a holiday schedule and that shops are closed, including grocery stores. Most school children will also have Friday off and many Austrians will take Friday off of work as well, making this a very popular four day weekend. This may mean that some things in both Salzburg and Vienna will be more crowded than usual over this period.
As for where to stay, when you say "budget is not an issue" I assume that a room over $250/night is out of the question, but please correct me if I am wrong. The best area to stay is the First District, which is the area inside of the Ring. The majority of sites are inside or adjacent to the Ring. Alternatively, you might enjoy staying in the 9th District (Alsergrund) which is where the university is located and this district is adjacent to the Ring, so not far at all from sites. Consider as well the 8th District (Josefstadt) and 7th District (Neubau), similarly well located, and really the coolest areas of Vienna. Lots of neat shops and restaurants. The 6th District is where you will find the major shopping street, Mariahilferstrasse. The 5th District, 4th District and 3rd District are also good, well located, especially closer to the Ring.
You will want to use public transportation in Vienna - it is cheap, fast and everywhere. Buy a 48 hour pass and you can hop on any form of transportation. Buy the pass online at https://www.wienerlinien.at/web/wl-en/24-48-72-hours-vienna . I also use Google maps public transportation function for getting around the city with great success.
I had a look at booking.com for your dates. I would recommend the following hotels which are available:
Hotel Indigo - very near the main market and walkable, but also very good public transportation. Work colleagues stay here routinely. Very new and modern.
Austria Trend Savoyen - The Belvedere Palace is in your backyard. Good breakfast. My parents frequently stay here and the Ring is just a few minutes down the road.
Magdas Hotel Vienna City - A social project, but very, very nice. Deliberately employs refugees and asylum seekers. Awesome garden and bar - very popular hangout spot. My parents have also stayed here and were very happy. Very happy little corner of Vienna quite close to everything.
With regards to getting off the train for a few hours on the way from Vienna to Salzburg, be careful that you have a ticket that allows for stop overs.