Green cars always require a reservation, I’m currently in Japan myself traveling with a green pass and from what we have notice, there is always free seats, even an hour before departure.
I can’t guarantee anything, as it all depends on the season, but shouldn’t be an issue. Atleast not for the moment.
As for the first day, the day of activation counts as day 1, regardless of when you activate it during the day I’m affraid.
Let me know if you have any other questions and hope it was of help.
With kind regards, Jonathan H.