Tokyo - Kyoto-Hakone - Tokyo

Hello,
My husband I will be arriving in Tokyo on the 24th of December and leave on the 7th of January. We were planning on being in Tokyo until the 30th of December then heading to Kyoto. When in Kyoto, we are wanting to take day trip to Nara (plus a couple of others) then head to Hakone on the 4th of January for 2 nights then go back to Tokyo on the 6th for one last night.

The 12 day JR pass is quote expensive so we were not sure if we could start the 7 day from the 30th of December, will it still be valid from our trip from Hakone to Tokyo on the 6th of January?

I am assuming we don't need the JR pass in Tokyo?

Your help/thoughts are much appreciated.

Thank you in advance.

Mem