Woodys Inn does Christmas dinners,It is on the road to Paralimni.Never been myself but my friends who live in Protaras say it's pretty good.The dafni pub just by the Odessa hotel stays open during the winter.

Protaras will more or less be closed down and it will be very quiet - only a few bars/restaurants stay open.

Protaras will be all but dead in December - guess you are staying at Constantinos?
Don't let this worry you too much though, I for one am jealous

We go regularly out of season (next time February) and love it, but for different reasons than those we go in Summer for. Winter can be a great time to find those little tavernas that stay open all year to serve the locals. We have found some of our favourite restaurants this way. Might be a good idea to hire a car though.

There is a thread over at Cyprus Eastern regarding Bars and Restaurants in Pernera (next village along) that is regularly updated on page 1 with regards to where is open/closed.
http://easterncyprus.com/viewtopic.php?t=1828