Christmas and Cycling in Cyprus

My partner (keen cyclist) and I are considering spending a week in Cyprus at Christmas 2012.

Is there any specific location or hotel that people would recommend which is cyclist-friendly - or maybe a good bike hire place ? Which areas of the island are best for cycling ?

At Christmas, is it likely that most of the historic sites / museums would still be open to tourists?

Are there other tours / trips /events that might occur around Christmas (or does everything close down in the same way as the UK does !)

Thanks. :-)