According to wiktionary's etymology, Austria is a Latinization of Ostarrihhi.
I see that au sometimes becomes o in Latin and daughter languages, e.g., Claudius -> Clodius, maurus -> moro.
Does o sometimes become au like the other way around?
By Latinic I assume you refer to the Romance languages.
AU > O is the typical result in Romance languages, although there are cases of preservation and some languages, like Occitan, tend to be more conservative in this regard.
In Catalan, dialectally, you can still hear people turning initial o's into aus initially. Not necessarily coming from AU. For instance, oliva being pronounced auliva. This is clearly not an etymological thing, but rather influence from some that are indeed etymological, like ocell/aucell 'bird', which comes from AVICELLU.