In many ways, Cooktown is still a frontier town and far enough away from Cairns to remain untouched by hordes of tourists. While it is not known as the fine dining capital of Queensland, it's still possible to track down a decent feed.
Probably the best options are the pubs. The Cooktown Hotel better known as the Top Pub has reasonably priced bistro meals and a great, lazy beer garden. The Endeavour Inn, on one of the main drags, Charlotte Street, is a colonial style building with a small bar and courtyard dining area. The menu features such exotic items as paperbark baked barramundi and kangaroo fillets.
The Sovereign Resort has a pleasant balcony restaurant overlooking the Endeavour River and although it's not the cheapest place in town, the food is high quality. For something novel, get aboard the MV Burragi , a refurbished former Sydney Harbour ferry now moored near the Cooktown jetty. It's licensed and BYO and the coral trout is a must.
If you're after takeaway, there is a handful of cafes that make OK toasted sangers, fish and chips, chicken and burgers.