In case the information in the link is confusing (it's a third party site not connected to Deutsche Post): the Post office is inside the "Press & Books" newsagents, Terminal 1, Level 0, near the regional train station. The opening hours for the Post office operations (9 am - 5 pm) are more restrictive than the general opening hours of the shop.

There are plenty of letter boxes thoughout the airport, but no Packstation (DHL parcel lockers) in the terminal itself. You'd have to go to Gateway Gardens or the International Mail Sorting Office (officially "Frankfurt Airport Building 190" - between Terminals 1 and 2) for that.