Marselisborg Palace is a summer home of the royal family, and when they’re not in residence the public can explore the English-style grounds and rose garden (free admission). When the blue-bloods are here, watch the changing of the guard at noon from a vantage point on the road. The palace is reached on bus 19.
The best way to check if the royals are in Aarhus is to ask around. Contact VisitAarhus, or check online for royal family activity (http://kongehuset.dk/en).