There are several bike shops in Aarhus where you can purchase a new bike. Prices will vary depending on your choice of bike.
If you are looking for a cheap, second hand bike, try to visit one of the police auctions, where left over bikes from the lost and found section are put up for sale. Police auctions in and around Aarhus are currently on a standby until August while the police looks for a new collaboration partner.
Another way to get a second hand bike is to check the websites http://www.guloggratis.dk/ and http://www.dba.dk/. Here you will find a lot of private persons selling second hand bikes. Many internationals leaving Denmark sell their bikes at favorable prices in the Facebook group Aarhus Internationals.