Does this mean, "We are going after her (in a queue)," or does it mean, (We are going after her (to catch her) "?
Or could it mean both, perhaps?
Either the former, or "we are following her". The latter would be אנחנו רודפים אחריה.
Anakhnu holchim akhareha
So this is used both for after in time and for after in place?
I heard "אחרה" (no yud) in a movie, at least I think that is how it would be spelled. It sounded like "acharah" and the English equivalent was "get her!" or "after her!" (guards).
Is "acharah" probably an imperative form of אחריה?
Prepositions, such as אחרי don't have "an imperative form", only verbs do. You probably heard אחריה "achareha" with a very faint "eh", which is how most people pronounce it.