Booking tickets is easy. I booked my Shinkansen Green Car ticket two months in advance, worried that the Mount Fuji view seats would be sold out during the cherry blossom season. I received the Shinkansen QR code one month before departure. On the departure date, just scan the QR code at the JR Shinkansen Station entrance and board.
Medium-sized suitcases (24") can be stored in the compartment above the seat. You can enjoy a view of Mount Fuji from the train window.