Hello, concerning the train between Budapest and Vienna: If you book the tickets a few weeks in advance on the website of ÖBB Austrian Railways you can get a cheaper rate - look for the SparSchiene fare: oebb.at/en/…sparschiene-ungarn. You can also compare with MAV (Hungarian Railways) www.mavcsoport.hu/en.
How long will you be going for? If 4 days or less, a return ticket costs €33 and you can buy at any time.
Ask on the Budapest forum about getting from Budapest airport to Vienna. There is a direct bus from Budapest airport to Vienna, but only once a day at 0525 www.flixbus.comEdited: 1:10 pm, October 17, 2017