Yeah, everything from Alpine to the Mogollon Rim will probably have significant snow in January. I'm not familiar with the Four Bar Mesa area at all, so I can't help you there. But I see it is over 6,000 ft, so it could potentially have some snow...not necessarily a bad thing depending on depth. If we end up with a mild winter, a couple of other areas worth considering are the Blue Range Primitive area, and the East Eagle Creek drainage just below the Rim, south and west of Strayhorse. Both of those areas hold Muleys as well as Coues, if that makes a difference.
If there is a lot of snow and you end up having to seek lower ground, there is access to lots of public land off of Eagle Creek Rd and Juan Miller Rd, both which you can pick up off the 191 between Four Bar Mesa and Morenci. Juan Miller actually goes all the way to the Blue River. If you need/want to get lower than that, you could try off of Martinez Ranch Road east of Clifton, north of Hwy 78. That country would be huntable regardless of weather, with plenty of places to glass from.