If you get through immigration and customs by 6:30 AM you could catch a direct train to Narita.

Otherwise I would probably do Keikyu to NEX at Shinagawa, as you suggested. You should be OK as long as you are out of Shinagawa no later than the 7:45 train. Keep in mind that you will run into rush hour chaos if it's a weekday morning, so I would be focused on getting through Shinagawa as early as possible.

I probably wouldn't do a taxi due to the risk of running into a traffic jam. The train is not risk-free but generally more reliable.