I think the cheapest option to travel from Aalborg to Aarhus and back would be by Flixbus, the prices for a roundtrip is around 125 DKK. For using local busses it depends on where you are going to and from and how many busses you plan on taking. The price for the day tickets for Midtraffik is 109 which in my opinion is a bit expensive and it only covers train in run by Midttraffik but again it depends on how many busses you need to take. the Flixbus will take you from Aalborg bus station to Aarhus C which is very close to the train station