Kansai International Airport (KIX) in Osaka is the major international airport in the area. It has direct flights daily from such major cities as San Francisco, Los Angeles, London, Paris, Amsterdam, Frankfurt, Helsinki, Dubai, Beijin, Shanghai, Hong Kong, Taipei, and Seoul (as of June 2018). Alternatively one can fly into airports in Tokyo (Haneda HND, or Narita NRT), and then on to KIX, or to Osaka Itami Airport (ITM) or Kobe Airport (UKB), which are domestic.
Do NOT confuse Kansai International Airport (KIX) with Osaka Itami Airport (ITM), whose official name is Osaka International Airport. The latter is currently a domestic airport, despite its name.
There are regular and frequent bus shuttles from Kansai International Airport to Nishinomiya and cities neaby (Osaka and Kobe). Osaka Airport Limousine will take you to Nishinomiya in 75 minutes, running 20 times a day, or to Umeda (central Osaka) in 70 minutes, running 60 times a day. Other options include shared shuttle taxis which take each passenger directly to their hotels. From Osaka Itami Airport, it will take 50 minutes to Nishinomiya using the Airport Limousine bus.
Those who fly into Tokyo can also reach Nishinomiya by Japan Railway’s famous and punctual “bullet trains” or Shinkansen. It takes two and a half hours from Tokyo to Shin-Osaka. Then you need to make an approprite transfer to other lines, depending on your hotel location.
Look at this page to avoid ticket troubles that often occur when you transfer from Shinkansen to other JR trains.
For information on Japan Rail Pass, look at this page.