Better run island, one in the Bay of Naples (5)
I believe the answer is:
Here is my best explanation:
'one in the bay of naples' is the definition.
The definition suggests a singular noun which matches the answer.
'better run island' is the subsidiary indication.
'better' becomes 'cap' (I've seen this in other clues).
'run' becomes 'r' (cricket abbreviation).
'island' becomes 'i'.
(Other definitions for capri that I've seen before include "Island off Italy", "I carp about the island", "Old Ford model - Italian island", "Romantic isle off Naples", "Island home of G. 24", "Mediterranean island", "Italian isle", "Romantic and expensive island in Bay of Naples", "Neapolitan isle" and "Island near Naples".)