The correct usage for 'on' and 'at'

Neither. You work at a place, at or on a task, but in a post. However, it would be much more idiomatic to say who works as a typist or who holds the post of typist.

A relevant definition of post is "place when on duty," 1590s, from Middle French poste "place where one is stationed," also, "station for post horses" (16c.), from Italian posto "post, station," from Vulgar Latin *postum, from Latin positum, neuter past participle of ponere "to place, to put" (see position (n.)). Earliest sense in English was military; meaning "job, position" is attested 1690s.

Online Etymology Dictionary

There is therefore strong connotation of place to post, which strongly justifies “working at a post” but not on it.

The use of on might be more weakly justified by the implication that someone working at a typing post would necessarily be working on tasks to do with typing.