Tokyo Hilton to Narita at 0500

I have a domestic flight on JetStar that leaves Narita Terminal 3 at 0800 Monday, 21 May 2018.
My family of 6 adults needs to get from Tokyo Hilton to Narita Term 3 by 0645.
Thus, we need to leave No later than 0530 by rail.
I see no rail options, however, that will get me from Shinjuku (or Tochomea Station or Nishi-Shinjuku station) to Narita before 0700. This is cutting it WAY close.
Am I correct?
Will I simply have to call a cab for 6 persons with 6 bags?

If so, what is the cost?

There’s buses running early in the morning that’ll get you to Narita in time for your flight. You may need to take a taxi to the bus stop, but the Tokyo Limousine bus might stop at your hotel.

There’s a mention of bus services in our tokyo Narita article:

Google maps won’t help you much with early trains connections but there are options.
Make two separate searches, shinjuku to nippori and nippori to narita, you’ll find that the first yamanote train is at 4.45 and you’ll get to Nippori in time for the 5.22 narita sky access which arrives at the airport at 6.12 total cost 1500 per person.
Limousine bus is more convenient, goes directly from shinjuku station but is 3000.

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