The question refers to the use of a Latin word, circa, which is sometimes used in written English to mean ‘approximately’ before dates.
The question regards “how well” the extension of this “works” in relation to temporal expressions like midnight.
I find this a curious way of expressing an idea. If it “works well” it will be in use in the language. Usage is Language! Experience and a variety of searches (such as this Google Books ngram) indicate that it is not in use.
My explanation of this restricted use is that circa was introduced into general English writing in the specific context of dates for compactness, especially were many are referred to in a historical context. No such requirement exists for “midnight” and the like. Further, I would regard it unnatural to use circa in speech, even for dates — compare: “We moved here circa 1990” to “We moved here in about 1980”.
Circa (or its abbreviations ca or c.) is used in specialized scientific contexts with measurements for similar reasons, although again its use in verbal presentations is discouraged (by me at least — compare saying et al. in a lecture).