There is a fishing boat that comes to the shore with the catch around 7:30 in the morning and you could ask there to buy fish. He will have though, what he caught. It is not a fish shop.
Your next possibility is a van that comes to Apollon with loudspeakers and he shouts that he has fish. He normally makes a stop in front of our house (not the studio. Where the apartments are), where there is a dead end. He parks shortly there, so that the people can buy fish, without stopping the traffic.
Both possibilities will work, if the weather is good. For example in August, there are high winds (meltemia), so you have less possibilities.
In any case in the port you always have the option to go and get some fish. There are lots of fish shops and in the supermarkets you will find both fresh and frozen.
Yannis and Stella @Apollon, Naxos 