Narita Express, validity period & use of JR Pass in Tokyo and Kyoto

Hi there!

Looks like a good trip! Here are some answers for you:

1. Yes, you will be able to activate the Japan Rail Pas at Narita and use it to travel into Tokyo on the Narita Express. Note that the activation office at Narita can get busy in peak times so you may have to wait a little.
2. You will be able to use the JR Pass for all of the destinations you list. The only exception is Hakone. For Hakone, the JR Pass will get you most of the way via the bullet train to Odawara. From Odawara you'll need to use the non-JR Odakyu lines upto Hakone itself. I recommend the Hakone Free Pass for this which can be purchased on arrival into Odawara.
3. You can use the JR Pass for all JR lines in Japan, but will not be able to use it on non-JR operated lines, such as the metro services. JR operates a really good service in Tokyo so you'll find you can get to most places via it (here's a map of JR in Tokyo). JR does operate a service within Kyoto however it is not as good as the metro or bus service. Expect to pay for metro or bus trips while you are there.
4. If you activate your rail pass on the 30th, the last day of use will be the 5th of Jan (30, 31, 1, 2, 3, 4, 5).
5. You can jump on all valid trains with the rail pass and use non-reserved seating including bullet trains. The only trains you'll need to make a free reservation for are the Narita Express which is reserved seating only.

Hope this helps!