If your session does get rained out, you should be able to exchange either for another session this year (and there are limits to this option), or get the same day/seats at next year's Open. You may have to be at the USTA's box office in Flushing Meadows to make a switch for this year's Open. And, there are exceptions depending upon how you obtained your tickets. Refer to the official page for detail. There are no refunds.
If your tickets are for an early session that gets rained out, then you have a chance of getting tickets for later this week. But you would have to be in Flushing Meadows right away to try and exchange at the box office and beat the other folks in the same boat. That probably means standing in the rain.
What If the Match Had Started Before It Was Cancelled?If a match in your session has been completed, you may be out of luck. It depends upon the time fo day (or night). See the official inclement weather policy for the nitty-gritty details.
What If The Finals Get Rained Out?"The Inclement Weather Policy does not apply to sessions #23 through #26." Ouch. That means officially you're out of luck. However, the US Open management would most likely honor tickets for a rescheduled final. They're just keeping their options open.
So You're Stuck in the RainYou're out in Flushing Meadows and it's raining. Either the match has been delayed, or cancelled, and you're wondering what to do. You're in luck! The US Open is one subway stop from the largest Chinatown in New York City.
Top choice has to be eating Sichuan food at Spicy & Tasty (39-07 Prince St, Flushing), a single subway stop to a singularly great taste. If the damp has dampened your spirits, there's nothing like Sichuan peppercorn to enliven. Try the dumplings swimming in sauce.