Senior Sergeant Mel Aitken said a member of the public reported the man had shot a possum off the pole in Caversham about 11pm on Monday and let a dog eat it.
The man was arrested without incident and a .22 calibre rifle seized.
He spent the night in custody and was summonsed to appear in the Dunedin District Court on February 12.
Police believe multiple reports of gunshots around Dunedin on Monday night most likely related to a car backfiring.
The reports were received from various suburbs including Kaikorai Valley, Mornington, South Dunedin and Port Chalmers between 11pm and midnight.
