To reach Spiti valley from Shimla, there are buses till Kaza, then from Kaza there are two buses to Shimla. If you are driving from sangla via Nako you reach Spiti. From what I enquired it takes three days to reach Spiti. Since you will be travelling on Hindustan Tibet highway don't miss out on the picturesque views. All through January there is snowfall kindly check if the roads are open to travel despite snow.
Reaching Spiti in January is possible only via Kinnuar Valley. There will be snow at the time so the transportation is completely dependent upon the road condition. But your best option is taxi a four wheel drive or something like an XUV. Buses do run but might be not as frequent.