Writing an exhaustive history of internationally known Austrian musical artists is embarrasingly easy.
If you narrow it down to those known outside of the German-speaking countries it is easier. If you narrow it down even more and only include people from Vienna it is even easier: There's only one left, since Salzburg has, perhaps unfairly, laid claim to Mozart[a].
He was born Johann Hölzel in 1957 in Vienna and died after colliding with a bus almost 41 years later. Inbetween those dates he released (among other tracks) Rock Me Amadeus, included here for reference, just in case you missed it in '85:
While Mozart seems to be the rope in a tug of war between Vienna and Salzburg with the former declaring 2006 the Vienna Mozart Year[c] while the latter presumably smugly noting that every year is a Mozart year in Salzburg, Falco doesn't quite get that amount of public attention. What he got was a stairway in the 5th district of Vienna, close to Ziegelofengasse 26 where he grew up:
The stairway was dedicated September 12, 2003, by the Vienna Chancellor of Culture Andreas Mailath-Pokorny with these words:
Falco hat das Lebensgefühl einer ganzen Generation in die Welt hinausgetragen. Er war der erste und wahrscheinlich einzige internationale Popstar Österreichs
Which in the interest of linguistic conformity translates to:
Falco took the attitude of a whole generation and brought it to the world. He was Austria's first and probably only international pop star.