Arizona weather is extremly dynamic. There is no way to tell for sure, use this link
http://www.wrh.noaa.gov/Flagstaff/ to keep track of what is going on in the area you want hunt. We are influenced by Pacific storms that move.
December and January can often be periods of really nice weather, but depending upon the elevation you can expect rain, snow and mud, or things could be hard, dry and warm.
4x4 vehicles or quads are pretty much necessary to get past rough spots, depending upon the area you are hunting in.
If you are into bow hunting, there is an OTC deer season in some units during the last half of December and through the entire month of January.