Going to Narita Airport very early morning

Which transpo is the best if we have to be in Narita airport (coming from Tokyo) not later than 6AM?
We're worried that trains and buses might not be open and running as early as 5AM. Would you know? I know there are car rentals and taxi as alternatives but we're worried that it might be so expensive.. Appreciate help. Thanks!

Buses and trains start arriving at the airport just after 6am. I strongly recommend to stay your last night at a hotel near the airport if you have an early departure.

I think there is no rntal car outlet that opens before 8am, so I guess that would not be an option either.

A taxi ride should be a possibility, but I recommend to make an advance reservation. The cost would be around 30,000 yen.

Staying at a hotel near Narita Airport will be the only option.

The transport options (i.e. trains) for early flights are designed to get you there on time from Tokyo for first departures.
For example, from Ueno the first Skyliner departs 5:40 a.m. arriving T1 6:24 a.m. which is good for a 9 a.m. international or an earlier domestic flight. Narita express is a bit later - earliest service departs Tokyo Station at 06:18 and arrives at Narita Airport at 07:18. There are bus options as well.
What time is your flight?

Interesting that no one is asking what part of Tokyo the OP is coming from, but with a "Family of 4", I too recommend checking in to an airport hotel later in the evening before your flight, or even earlier to explore Narita.