As a result, the final started 40 minutes late and there were scenes of violence inside the stadium with a firefighter injured by fireworks thrown from the stands.
The shootings occurred in an area where Napoli fans were gathering for the match, the ANSA news agency reported.
The fan in serious condition was reportedly shot in the chest, while the police officer was shot in the hand.
Several other fans were also reportedly injured.
Circumstances of the shootings were not immediately clear.
Disturbances were also reported outside the stadium.