How did Spider-Man do it?

First of all, remember that it was never planned and most of it was improvised. From screenrant:

Appearing at a high school in Iowa, Joe Russo revealed some cool new information from the latest Marvel offering. While he won’t share any spoilers for Infinity War and any future MCU films, he shared that Holland’s heartbreaking plea to Tony Stark was improvised. Courtesy of Reddit user Anewthrowaway_quest, the filmmaker shared he just instructed the young actor to act like he doesn’t want to leave, it was Holland who decided to ad-lib his lines: “Mr. Stark, I don’t feel good” and the now iconic “I don’t want to go” phrase. It’s unclear if RDJ’s dialog telling Peter he’s fine was also unplanned, but considering the veteran’s knack for improvisation, it’s not impossible.

So it does become little more stretched than other deaths but also remember he does have a sixth sense called "Spider-sense". So we can presume due to it he felt that something is coming and he did see that people are disappearing so he knows what can happen to him.

Even remember his premonition is not fully consistent in comics either, even one of his clone had it so strong that he could even see future:

Kaine’s Spider-Sense is more powerful. He can actually see the future, not just get an idea something is about to happen in the next moment. He can’t control his power and choose what to see. Rather, he gets visions of events to come. For instance, in “The Spectacular Spider-Man” #222, by Tom DeFalco and Sal Buscema, Kaine is in the middle of sorting clones from the true Peter Parker when he gets a vision of Mary Jane’s death in the future.- cbr

So maybe MCU Spider-Man's premonition/spider-sense is strong enough that he can sense that his reality is getting altered. He was able to sense alien ships from far, so maybe he can sense reality-altering events too.

This is what Weta VFX’s lead supervisor, Mat Aiken say about it:

“Spidey’s really fighting it. He definitely doesn’t want to go and he’s fighting… he’s incredibly powerful, so he’s able to withhold it for a lot longer than some of the others, but in the end he can’t withstand it.”