Shinagawa station is not too complicated. 17 minutes is plenty as AlwaysAisle says. But there is always a possibility, albeit unlikely, that the bullet train runs a bit late and you miss your scheduled Narita Express. So I would hope that the next Narita Express after that will still get you to NRT in time for your flight.

I don't know how often Tokaido shinkansen runs late. Probably not often. But other JR limited express trains like Azusa run late all the time. When I have a plane to catch, I try to give myself >30min for Azusa>NEx at Shinjuku.

The transfer time at Shinagawa from the bullet train also is impacted by which car you're sitting on the arriving bullet train. Depending on your car, it might involve a short or long walk on the platform. One can specifically ask to be assigned to a car convenient for Shinagawa transfer. However, in your case, your green car ticket will pretty much dictate that.