The beach strip running from Agios Prokopios south through Agia Anna has accumulated enough tavernas and beach bars to constitute a genuine eating option rather than a last resort. Lunch here is uncomplicated: fish fresh from the harbour at Agia Anna, Naxian salads, cold beer, sea in front. The quality of the raw ingredients on Naxos means that a simple meal done without pretension is usually satisfying, and the competition between beach establishments keeps standards from slipping too far.
A few of the tavernas at Agia Anna have been operating long enough to have developed a local following — the reliable sign that the kitchen is actually cooking rather than defrosting. Look for the tables that Greeks are sitting at rather than tourists; the distinction is usually visible and the implications reliable. Fresh fish priced by the kilo is always worth asking about: the morning's catch at the Agia Anna harbour fishing boats determines what's genuinely seasonal that day.
For the beach itself, several establishments rent sunbeds with the understanding that you'll order something during the day — a drink minimum at the lower end, a meal at the more organised operations. This is the standard Cycladic beach arrangement and entirely reasonable. Order a long cold drink around midday, proper food by early afternoon, and leave before the evening prices adjust upward.