I'd just fly from Narita. You are already at the airport, the airport in Fukuoka is in the middle of the city and super convenient on the subway.

13:20 should be quite doable, Jetstar checkin closes 30 minutes before departure. But that is in T3, the transfer counter in T2 will close 60 minutes before.

If you do take the Shinkansen, yes indeed, as mentioned, change at Shinagawa. Tokyo station has the Narita Express in basement level 5 and the Shinkansen a level or two above ground. Where Shinagawa has the two platforms pretty close together.