What you want to do in Flensburg around Christmas is beyond me. I would hit Hamburg or Lübeck instead, or if you really want to see the real north, hang out on one of the North Sea islands.
I took a quick look and apparently there is only one hostel in the area around Flensburg, the DJH Nordmark.
http://www.djh-nordmark.de/index.php?id=411
This is an official HI hostel and in Germany these cater mainly to school classes and youth groups. I'm not surprised it is closed during Christmas, as Christmas is the really the lowest season for the vacation spots on the coast.
You should go to the official tourism site http://www.flensburg-tourismus.de/ and look for private rooms or pensions. (The German version of this site has a room search option, it is on the right hand column under the telephone number of the info centre.). You can also try to email the tourist information, adress is info(at)flensburg-tourismus.de