I was told that all the trains which depart from Switzerland for
mountain tours and even some destinations reachable from switzerland
are do-able from Austria as well
Well of course it's doable, if you have all the time in the world and don't mind a lot of train changes. But if you're asking about day trips, you'll have to be more specific. What places do you want to see? And why wouldn't you stay near the sites you want to see? I'm assuming you're asking about hotel stays, and not actually living in Austria? Or are you planning to move there?
ETA. I've just had a look at several of your other posts that you made today, and I have to ask- have you looked at any guidebooks? And, as Fast Eddie mentioned- maps? I really don't think you've given any consideration to the distances you're planning to travel, or the time it will take. Personally, your itinerary is one I wouldn't touch with a 10 foot pole. You need to do a lot more research into each of the places you want to visit, and decide how much time you will need in each place to see what you want to see. And spend some time on the appropriate websites looking at travel times and connections required. So far, more than half of your time seems to be spent on trains.