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.
Click for more details and booking links Cheap Flights: Austin or San Antonio to Tokyo (HND/NRT) Japan $993-$999 r/t [August-April] – American Airlines