Rule about capitalization of titles: capitalize only when used before the person’s name or as an address: You’re the king, Your Majesty.
The outgoing president of the United States has written a letter to the newly inaugurated president. Instead of friendly advice, the letter contains the horrible truth that the public doesn’t know.
The first sentence is somewhat clunky, but I can’t think of a quick, minimally invasive way to fix it. It probably needs to be completely rewritten, something along the lines of The newly inaugurated president of the United States receives a letter written by her predecessor.