JR Pass Activation and Ticket Purchase

I have a 7 day JR Pass (First Class Green Car) and will be traveling to Japan and arriving in Tokyo December 27th. I would like to travel to Kyoto on Monday December 30th and return to Tokyo from Kyoto January 4th.

When I arrive on the December 27th and go to a JR Pass location to exchange the voucher with an activation date of December 30th can I purchase at that time tickets to Kyoto from Tokyo on December 30th and a return trip from Kyoto on January 4th?

I know the regular travel cars and reserved tickets are already selling out, will the green cars see more selling out as we get closer?

I know travel is very busy due to the holidays, my daughter is in Kyoto and I want to make sure I can arrange travel to visit her.

Thanks Mark