Is january the best time to climb mt taishan? In terms of visitors tourist traffic, weather etc

To be honest, April to November is regarded as the approproate time to visit this mountain.
About January, this month is not the tourist peak season. There won't have crowded people. So don't worry.
About the weather, it was 10 F to 20 F according to the previous weather statistics. It will be cold and dry. You need to prepare heavy overcoats, jackets, sweaters and pants in order to keep warm.