The Nissan Micra left the road and hit the postbox in the early hours of Thursday morning (September 16). A passing motorist called the emergency services.
The cast iron postbox was smashed to pieces and thrown across a wide area, including in the entrance to Thorpe Hall.
Both airbags in the car had deployed and the front of the vehicle had sustained significant damage.
Police, ambulance and fire crews responded to the incident.
A police spokesperson said: "We were called at 1.42am this morning (Thursday) with reports of a collision in Thorpe Road, Peterborough, where the driver fled the scene.
"A 20-year-old man from Peterborough was arrested nearby on suspicion of drink driving, taking a vehicle without the owner’s consent, failing to stop at the scene of a collision and failing to report a collision.
"He’s currently in custody at Thorpe Wood Police Station."
A spokesperson for Cambridgeshire Fire and Rescue said: "Crews were called to a road traffic collision at around 1.45am on Thorpe Road. A vehicle had hit a post box.
"No action was required from our crews and they returned to station by 2.10am."
No one was injured in the incident.