Monument Valley is high desert, it has seasons, the weather will be what it will be in winter. Roads are usually passable. Prepare for windy, subfreezing condtions for your outdoor activities. Not unusual for parts of the area you'll travel to accumulate a foot or two of snow by January, depends on what side of the mountain relative to direction of storm travel.
The site is on reservation land, limited commercial development. Visitor accommodations are at trading posts, the post for Monument Valley is Goulding's. I think it is pretty nice.
The valley is not far, by local standards for driving times, from the small cities north of the reservation (Mexican Hat, primarily) which offer more amenities and choices than a trading post. For that matter, Monument Valley is not all that far from Moab, which is the center for high-desert red-rock tourism and recreation in southeast Utah. It is further, but still less than a half day's drive, from Farmington, New Mexico, the 'big city' in the area.
Your direction of travel, last 'big city' is Flagstaff, last place before the reservation with multiple accommodations could be Winslow, other places to see include Grand Canyon, red rock tourist area of Prescott-Sedona. Sites in SE Utah, SW Colorado are beyond Monument Valley. There are more canyonland national parks in SW Utah as well.