I hope to get an answer to my letter by ____ of post.
Note: I checked “next of post”, “return of post”, “back of post” and “round of post,” it looks like only “return of post” exists as a set expression. I also checked dictionary, this is what I found:
by return of post -
(British) by the next post back to the sender
Does “post” here mean “delivery”? Can the original sentence in the question be interpreted as “I hope to get a response in the mail by the next round of delivery”? If not, I'd appreciate your correction. Thanks