The good news about Aarhus is, despite its relatively large size, many of the cultural, nightlife, and cuisine highlights occur within decent proximity to each other, making it easy to satisfy the best of all three criteria in day's time. With classic pedestrian squares like Store Torv and fantastic examples of antiquity (Aarhus Cathedral), the area just north of the river is an obvious cultural draw. Even better, you'll find a great selection of restaurants and nightlife for when your interest demands it.
Three streets in particular are valuable on both fronts: Klostergade, Vestergade, and Skolegade. The first two run parallel to each other north of the Aarhus River, forming a rough vicinity of cafes, take-out joints, and bars at the center of which is Klostertorvet. A few minutes of walking (or biking) in the area yields some moderately cheap options for both dining and drinking, including popular options like Café Smagløs (great coffee and brunch), and Social Club (crowded and super-hip dance vibes).
Further east, Skolegade runs north and south for only about four blocks, but still packs a punch with its late night options. From rock clubs (Fatter Eskil) to smooth cocktail lounges (Sidebar) and sweet and sweaty dives (Escobar), your bases are covered on Skolegade. You can even catch a performance at the Aarhus Theater at the street's northern end.