Is the temperature unbearably cold in December through to January in Shanghai?

No, this city doesn't have chilly cold winter. Generally, the temperature may range from 40 F to 50 F. You are suggested to prepare overcoats, sweaters, long johns, pants and scarves. :)

When we were in Beijing, the lowest it gets was -8 degrees. I hardly feel the cold, we even have to sleep with the windows open because the hotel aircond were slightly too warm for comfort. SH winter would be like a cross between Canberra/Snowy Mountains in NSW. If you can dress appropriately while Snowy Mountain regions, you will be fine there.