I am not a teacher.
I believe "My cat went missing ON Friday" is correct.
You may reference Purdue University's Online Writing Lab (OWL) by searching "Prepositions for Time, Place, and Introducing Objects." Sorry, at this time I am not able to provide a link.
As for the "went missing" part of your sentence, I find it too colloquial (too much like slang). However, as Curmudgeon pointed out, it is more common and accepted in Britain, and it is often used in the American press.
You might try "My cat ran off (away) on Friday," "My cat was missing on Friday," or "My cat has been missing since Friday."