the nearest international airport to tianmen cave will have to be guangzhou baiyun international airport. you can then either take a train to zhangjiajie city or a domestic flight from guangzhou. when in zhangjiajie city, you will surely be bugged by numerous touts, however, you can just take a cable car up to tianmen cave. the cable car station is right at the city quite near the train station, you can’t miss it. unfortunately, you’ll have to be part of the military to be able to fly a plane through the cave.

i hope this helps.

thanks for visiting!