Post as in postal service seems to come from the Latin “positus”, meaning “placed” or “stationed”. Dixit Perplexity:

This refers to the system of relay stations or “posts” set up along routes to deliver messages and mail. In ancient times, postal systems relied on a series of stations where horses and riders could be changed to speed up long-distance communication. These stations were called “posts”.

The term “post” originally referred to the system and infrastructure for delivering mail, rather than the mail itself. It came to mean both the system and the items delivered over time.

Post as in postpone is a Latin preposition meaning ‘after’ or ‘later’.