om = bărbat, not just ”human”? Wiktionary says that the use of om as an alternative to bărbat is rare today, but perhaps the course makers want us to learn that we might encounter it in old books etc?
If somebody would say that sentence to me in everyday life, I would assume it's about a man (specifically, an adult man). Otherwise, I'd expect them to say "O să apară o femeie..." or something similar.
But would it sound old fashioned? Maybe there are dialectal differences in bărbat vs om?
No, I don't think it would sound old-fashioned. Not to me, at least. (I'm in my early 30's btw...) :-P