If you have a direct link to an old flight on flight aware, it is still available for public viewing. It is true that for an enterprise subscriber looking at the history for the flight, you'll only see eight months back. So you just need to create your own link.

Here is how I found this flight using this link to start: https://flightaware.com/live/flight/...510Z/KDTW/RJTT

Date: Changed to 20170903
Time: The time is the originally scheduled departure time of the flight in zulu/UTC time. Can take some trial and error if you aren't sure of the departure time. In this case the flight was at 12:10 EDT which is 16:10 UTC. Changed to 1610Z.
Airports: The airport code is in ICAO format, changed to RJAA which is the ICAO code for Narita airport.

With these changes we end up with this: https://flightaware.com/live/flight/...610Z/KDTW/RJAA

Finding the registration for a domestic flight is very easy in comparison, the Bureau of Transportation Statistics has a great search page: https://www.transtats.bts.gov/ontime/Departures.aspx