Have ceiling damage in daughter’s college condo shower area..can I fix?
I think your plan is fine, assuming there isn't a persistent plumbing issue. I'd gently brush and scrape away just the loose material, striving to retain as much texture as possible. If it gets smooth, dab on some spackle or joint compound to replicate the texture, then prime and paint. 

Recognize that your paint probably won't match the existing very well, so keep it small and feather it in.