Hi there!
Traveling around the new year holidays can be somewhat chaotic in Japan. As reserving seats for the period 29 December - 2nd of January is certainly recommended. You can start making reservations once you are in Japan, to do so, go to the ticket office any JR Station, show your JR Pass and tell them staff when and where you would like to travel.
Opting for a Green Pass, is a great option although not necessary. Trains between Tokyo - Osaka have a good amount of Green Seats, thus it may be easier to find a seat. The real reason to consider going green however, is that you can skip some of the crowds and have a much more relaxed journey.
Have a pleasant journey!
Daniel