1650s, "public department in charge of letter-carrying," from post (n.3) + office. Meaning "building where postal business is carried on, office or place where letters are received for transmission," is from 1650s. In slang or euphemistic sense of "a sexual game" it refers to an actual parlor game first attested early 1850s in which pretend "letters" were paid for by kisses.
1761, "any disabled person or thing;" especially Stock Exchange slang for "defaulter."
A lame duck is a man who cannot pay his differences, and is said to waddle off. [Thomas Love Peacock, "Gryll Grange," 1861]
Sometimes also in naval use for "an old, slow ship." Modern sense of "public official serving out term after an election" is recorded by 1863, American English. The quote attributed to President Lincoln ("[A] senator or representative out of business is a sort of lame duck. He has to be provided for") is from an anecdote of 1878.
It is well known to everybody who knows anything of its history, that this court [Court of Claims] was made a sort of retreat for lame duck politicians that got wounded and had to retreat before the face of popular condemnation. That is just exactly what it was for, a safe retreat for lame ducks; and it was so filled up; (etc.) [Sen. John P. Hale, New Hampshire, Congressional Globe, Jan. 12, 1863, p.271]