It's technically correct, without being exactly colloquial and just generally, 'abreast' used like that can lend itself to some puns on mammary lines.
Also, 'abreast' is a word that relates to news, current events and professional matters (eg the latest orthodontic techniques), rather more than say, personal matters. 'Abreast' suggests that the events concerning Person B are in the public arena, in the Kardashian sense.
It would be safer to say "he brought me up to date", "he let me know" or even "he updated me" - depends on how sensitive the information is.
If you are talking to a third person, not person B, you have rather more colourful options to use, such as "marked my card" or "had a word in my ear" (you will even hear, "had a word in my shell-like" (= ear). These are a bit gossipy.