The bar scene in Wolfenbüttel is quiet and local. A few bars and Kneipen around the Stadtmarkt serve the usual German beer selection alongside, inevitably, Jägermeister in various forms. This is the hometown of the stuff, after all, and local bars sometimes carry limited editions not available elsewhere.
For coffee and cake, the cafes on and around the market square are perfectly pleasant for a mid-morning or afternoon break. Expect traditional German Kaffee und Kuchen rather than specialty coffee culture — bring a book or enjoy the view of the half-timbered facades.