This has been mentioned before: wait until you get to Japan. Unless you're traveling during a holiday period, you're over-engineering the situation. Consider that you can do it as soon as you arrive, so that would probably be at least a few days if not longer before your departure day. It is unlikely that the green car will sell out even a few hours before departure, let alone days in advance. Plus, they leave as frequently as every 5 minutes, so there is a large capacity. You can go to any JR ticket office anywhere in the country and make a reservation and buy your tickets. Just walk in and buy your ticket from a person at the counter. Simple.