China Eastern Airlines and Korean Air are operating from Shanghai and Seoul, but these flights are not served daily. You should check the flight schedule beforehand if you are to use these international flights.
On the other hand, there are many domestic flighets to Nagasaki every day. You can fly directly from Tokyo Haneda Airport (by JAL, ANA and SNA), Osaka Itami Airport (by JAL and ANA), Nagoya Centrair Airport (by ANA), Nagoya Komaki Airport (by JAL) and Okinawa Naha Airport (by ANA). Oriental Air Bridge, a local airline of Nagasaki, is operating the flight from Miyazaki, Kagoshima and neighboring island -Tsushima, Gotou and Iki- , with prop jet.
JR railway company provides express service from adjacent prefectures. From Hakata Station in Fukuoka or Tosu Station in Saga, you can use the Limited Express "KAMOME"(to Nagasaki), "MIDORI"(to Sasebo) or "Huis Ten Bosch"(to Huistenbosch Resort in Sasebo).