There is another option: especially for travellers arriving after a very long flight, why not travel from the airport to the city, spend one night in a hotel to catch up on sleep before travelling further?

This way, you could book a very cheap fare for the day after arrival and get on the train well rested. The cost of the one night in the hotel could be balanced by the savings in the train fare, but you'll be rested and the chances of missing the train are very small.