The short answer is you just walk up and buy the Shinkansen tickets when you are ready to travel, unless it is during the very busy New Year or cherry blossom season.
I tried to figure out how to buy tickets online for our November trip ( 2 months ago) and after reading a lot I concluded it was best to just buy them in Japan. We had only one journey ( Tokyo to Kyoto) so we were not using a JR Pass. There is no discount for buying tickets ahead.
But I am so used to buying our train tickets in advance that I thought we should at least get the tickets at the station the day before our travel. We stood in two different lines for Shinkansen tickets, only to be told when we reached the window that they only sold "same day" tickets. We were directed to a different ticket window "outside" but in the huge Tokyo Shinagawa station we could not find it. So we decided to just show up and get our tickets the next day when we were ready to travel.
Turns out that was the way to do it. We had a leisurely morning, going for a walk to explore a bit, before checking out of our hotel and walking to the station. There was barely any line for tickets, and the very helpful agent sold us tickets on a Nozomi train departing 30 minutes later, with seat reservations on the "Fuji-san view side" ( the right side if traveling Tokyo to Kyoto).
30 minutes was just about right for us to head to the platform, walk down and find our car number, and line up in the proper spot. The platform is marked with car and seat numbers so you know which end of the car to board. Lines painted on the platform keep the queue in place. It is all very orderly and efficient.
If you have more than one trip by Shinkansen you should look into the JR Pass, but we did not, so I cannot help with that.