Kansai International Airport has already determined to halt its operation on September 5th.(Some airlines expect it will prolong until Semtember 6th)
Anyway, unless transportation of the bridge connecting the airport and mainland resumes operation, the airport wouldn't be able to recover its functions.
Railways are gradually resuming their operations. Here are information as of 0:30, September 5th.
・List of suspended rail lines on major private railways
Kintetsu:Osaka Line between Osaka-Uehommachi and Yamato-Yagi
Ise-Shima Line between Uji-Yamada and Toba
Minami-Osaka Line, Nagano Line, Gose Line, and Yoshino Line
Hankyu:Kyoto Line, Senri Line, and Arashiyama Line
Hanshin:No line is suspended
Nankai:Airport Line between Kansai Airport and Rinku Town
Koya Line between Koyashita and Koyasan
Kada Line, Shiomibashi Line,
Keihan:No line is suspended
JR lines in Kansai Urban Network aren't updated. But, they will resume operation except for Kansai Airport Line.