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).