Old Glory can be found on the body of Ulysses, if the player chooses to fight him.
It appears in Ulysses' footlocker upon the completion of Lonesome Road.
Old Glory was originally to be a weapon in Fallout: New Vegas, then named the "Eagle Pole". It was cut from the base version of the game as a usable item, but its visual resources remained intact.
The 3 of Clubs in the Fallout: New Vegas collector's edition card deck depicts Ulysses holding what appears to be a flag pole.
It is possible to get two copies of the weapon; if you finish The End with Ulysses still alive he will be sitting on a rock ledge, over looking the The Divide after you return back from the Mojave Wasteland. Alternatively you can just kill Ulysses before completing The End/The Apocalypse, take it from Ulysses' corpse and another one will be in the footlocker waiting for you.
Ulysses uses a unique non-player character only version of this weapon, which cannot be disarmed from him or looted from his corpse. There are 4 different versions, depending on the player's level (increasing in power at player levels 25, 35, and 45) making it the only weapon in the game that is scaled to the players level. Each also has a 10% chance to knock the player down (except against Stonewall).
The idea of carrying a gold or silver pole with an eagle on top dates back to the Legions of the Roman Republic in which each legion had two "Eagles" which were to never be captured. They symbolized the courage of the members of the legion. Since Ulysses was/is a member of the Legion, he may have been an eagle bearer, showing Caesar's insistence on imitating Rome.
Ulysses' dialogue in Lonesome Road indicates his eagle staff is an Old World symbol.