Mount Tai is open all year round. Be careful not to go on snowy days. Take some clothes. It will be very hot when going up the mountain, but it will be very cold when resting.
If you want to go up the mountain in the morning and down the mountain in the afternoon, I suggest you take a bus to Zhongtianmen, then walk up the mountain and take a cableway down the mountain. There is plenty of time.