I didn't find a duplicate, although "ROC to TYO - how can I reduce the cost of my flight?" was helpful. My son was accepted into a training program at the south end of Japan. He was advised to fly to Narita (Tokyo) and take a shuttle all the way down there. I found that flights to Osaka (KIX) are typically cheaper, but I'm not sure of a way to find whether another city is lower still. All the airports I checked near his destination were much more expensive than Tokyo.
There are only four airports in Japan that have direct flights to the continental US: Narita/NRT and Haneda/HND (Tokyo), Centrair/NGO (Nagoya) and Kansai/KIX (Osaka). None are particularly close to Kyushu, and it will probably cost you well over ¥10,000 in train fares to get to your destination, unless you're going to slum it in local trains that take forever (not recommended).
- If you fly in with any oneworld airline, you can get any connecting Japan Airlines domestic flight for ¥10,000 (~US$100) with the Yokoso Japan fare. The same deal applies to Star Alliance carriers and ANA domestic flights with their Visit Japan fare. These fares never show up in online booking engines, you need to book via a travel agent or the airline directly.
- For major routes like Haneda-Fukuoka, Japan's low-cost carriers may offer even cheaper fares if you book in advance. Check out Peach, Vanilla and StarFlyer. Note that the LCCs only fly to big airports, while you can use Yokoso/Visit Japan fares to go directly to any airport, no matter how tiny.
- While it has no direct flights to the US, Fukuoka Airport (FUK) at the northern tip of Kyushu has good connections to Asia, so it might be possible to score a good fare transiting via eg. China or Korea to Fukuoka.
- Last but not least, Rome2Rio is really good at digging up obscure flight combos. Trying New York to Beppu, it suggests flying from NRT to Oita with Jetstar Japan (GK), or Asiana via Seoul to Kumamoto.
- And skyscanner.net like Greg Hewgill said: Put in "USA to Japan" and narrow it down from that.