I am planning on driving from Tokyo to Nagano station and then taking the bus to the snow monkey park.
How safe is it to drive to Nagano from Tokyo at this time in January?
Barring a major winter storm, driving from Tokyo to Nagano is very safe. The highways are well-maintained and easy to drive on (but do require winter tires during the winter season). Nagano City does not get much snow either, so the drive to the station should be smooth.
I hope that answers your question!