The Nuwara Wewa Guesthouse has quite nice, though somewhat westernized meals. Non-guests can eat at the restaurant. There are numerous other similar guest houses, and the Tissa Wewa guesthouse gets good reviews.
There are many hotels Sri Lankan English for a small restaurant in the Anuradhapura New Town, which offer an authentic Sri Lankan meal, at quite a good price. Smaller boutiques offer quick eats, such as pastries and rolls, which are also quite good and cheap. Stores and small markets in the New Town and surrounding area often carry snack food such as chips, chocolate bars and nuts, as well as cold drinks. Some also have fresh baking, and many markets sell fresh fruit. Lunch packets, generally sold between 11:00 a.m. and 2:00 p.m. are an excellent lunch option. Unless you are a fan of spices, ask the server to not put in too many chiles when possible, as authentic Sri Lankan food is very spicy.
Cargills Food City is a small grocery store which carries many Western products, as well as fresh produce and meat. You can get all the essentials for preparing your own meals here.