I think he can make it. He need around 20 hours from Japan until he make it to the prison, so round trip is around 40 hours. Assume the dinner is on 5-6pm (cause winter night is faster) he has 7 hrs spare to tell the story and wait for the snow to stop. Remember that it's "the day after tomorrow". The reason kana's phone call didn't reach because he's already on plane, and rushing to japan.
In this story time isn't aligned properly on purpose. So touma visiting the prison is happening before kana shopping for dinner.
@Miyacchi Yes. That's why she is only on silhouette. It's to hid the fact that caff is already an old man, because caff is still on his time 40yrs ago.