No hike at Shinagawa from Shinkansen to NEX or vice versa, its pretty much easy and strait forward at Shinagawa:
You can see at my site:
http://www.tokyosubway.info/shinagawa.html
Tokyo Station on the otherhand requires a bit of a transfer:
http://www.tokyosubway.info/shinkansen.html
(Just follow it backwards)
Either case 20 minutes is enough time to do it at either, Shinagawa can be done in a few minutes, Tokyo a little longer, because of the distance, but both doable in the amount of time. I have a preference for Shinagawa because it's nicely laid out.
The JR Ticket office at Kyoto Station, can reserve all the way through for you, actually, any JR reservation office at any JR station can do it for you as well.