When this post is automatically published I shall, hopefully, be on my way to Austria for a holiday. Despite the names Austria and Australia being similar, the Aust of Austria means East whereas the Austral of Australia means South.
Austria is the Latinised pronunciation of the German word for Austria, Österreich, which looks as if it translates literally 'eastern empire'. In the tenth century the name was Osterrichi, which meant 'eastern province' or 'eastern marches'; the territory at the time was on the eastern edges of the Margravate of the Babenbergs. The Margravate became a duchy in the following century and eventually became the nucleus of the Habsburg empire. Aust is not that different in pronunciation from the German öst.
The name Austria is completely unrelated etymologically to Australia. The latter's name is from the Latin australis, meaning southern.
I'll be away for about nine days, but there are a couple of posts programmed to be published over this time. I won't be responding to comments though, so apologies for that in advance.