According to the excellent book Ghost Sightings by Brian Innes, the oldest written report of a ghost comes from the Bible, in the first book of Samuel. Saul goes to a medium (“a woman that hath a familiar spirit”) and asks her to conjure up the deceased Samuel, which she does. Samuel appears in the form of “an old man covered with a mantle.”
Another very old ghost sighting comes from Ancient Greece. A Greek writer named Pausanius wrote around 150 AD about a haunting at the site of the battle of Marathon (490 BC). In the words of Pausanius:
“At this place you can hear all night horses whinnying and men fighting. No one who stays there just to have this experience gets any good out of it, but the ghosts do not get angry with anyone who happens to be there against his will.”
Notes: The description of Samuel in his undead state, covered with his mantle, seems to set a precedent for the sheet-covered ghost so favored in Western culture. Although Samuel was conjured and perhaps not a “true” ghost, Pausanius’ account is clearly that of an already established haunting which must have been well known at the time.