Learn English – Washing the skin of a dead rat

To stay one's hunger is perfectly acceptable, and not at all uncommon (though it's actually more likely to be "stay your appetite"). Less common, and with less of a "temporarily" sense, would be assuage one's hunger. But I think probably the most common idiomatic usage is...

Here - have a sandwich to stave off your hunger until dinnertime.

...which my mother used to say to me when I was a child. Perhaps this one gains traction by alliteration / association with both stay and starve.

to stave off
to fend off, to ward off (something adverse) _{Merriam-Webster}

(To metaphorically repel / drive away using a stave = staff = stick.)

It's a brilliant pun. Hit rewind is reminiscent of cassettes (video or music); it can also be used euphemistically to mean "go back in time."

It's not a commonly-used idiom, but it's well-understood in most contexts.

I looked through some Google Book results. The modest numbers indicate the phrase is not all that commonly used. More often than not, it refers to the physical act of hitting a REWIND button on a tape device:

"He'd brought the tape in, so he felt personally responsible for it. McGrath hit rewind again and tried once more." (L. Child)

"Then he remembered that he'd forgotten to rewind the tape, so he hit rewind and the tape flew backward on the reels." (R. Birch)

But some metaphoric uses of the phrase are also sprinkled in the results:

"We thought, if someone somewhere could just hit rewind then maybe we could start over. But we couldn't. We can't." (J. Stowe, writing about the Space Shuttle Challenger explosion)

"Wish I could go back, press pause, freeze time, then hit rewind, and right before he grabbed that knife, hit delete. Then I wouldn't have been afraid to go to sleep. (A. Russell)

Since the NY Times writer you cited is talking about using an old technology – one that is becoming antiquated, but happens to have a rewind button – it's a very apt usage of the phrase.