American Airlines has round-trip flights from Austin or San Antonio (AUS, SAT) to Tokyo (HND/NRT) for $993-$999 with a connection. Flights August to April. Departures on Monday, Tuesday, Wednesday, and Sunday.
Notes: Haneda is closer to Tokyo Metropolitan area than Narita.
Exchange rate is greatly in favor of USD/JPY. If you stick to Japanese domestic goods, you should be able to take advantage of the favorable exchange rate.
A Carry-on bag and also a personal item is allowed for free on American. You can avoid paying for seat selection by accepting the randomly assigned seat at the gate.