Have you tried the Japan Taxi app? It's slow as molasses to start up but it has an Uber-like interface and supposedly works nationwide; in Tokyo it's served by Nihon Kotsu.

For NRT I've done cab to Ueno then hop on the Skyliner -- I learned my lesson about not taking N'EX during inclement weather, last year when a lightning struck the overhead cables near Sakura, causing the N'EX to go out of service. For HND I usually cab it, so I can sympathize with OP. (Note: this is all business travel so I'm not paying in cash, but in time)