Even though there are few airports in Denmark (two of them are international) we highly recommend to fly to Copenhagen Airport because the train connections are faster and more transparent.
At Copenhagen Airport you NEED to buy the train ticket to Odense. The train takes approximately 1 hour and 45 minutes and it first leads to Copenhagen Central Station, where you need to change track and take the train to Odense.
Check the trains here: www.dsb.dk
*sometimes they are direct, otherwise take train to Frederecia, Esbjerg or Aalborg, where you have Odense as the intermediate station*
From the Odense Train Station (Odense Banegård) to the apartment the transport will be arranged.