I think it became common knowledge that Fury was alive after the climax of Avengers: Age of Ultron. At least, SHIELD was likely aware and the Sokovia Accords made that information known to other major Government Bureaucracies (FBI, CIA, DODC, SWORD).

Even if Coulson tried to keep it on the down low, other Agents loyal to the Accords would have been obligated to tell Mace and Talbot. Pretty much all agents would have known after the reveal of Project Lazarus.

Either Nick Fury being alive was leaked to the public in Winter Soldier (unlikely given what is presented in the end of the movie), or he started, maybe, having contacts with more important figures over the years, revealing he is alive, or people just told he was, maybe the Avengers did, during the years he was snapped.

My guess is the second option because, in Infinity War, we see Nick Fury out in the open in a street with Maria Hill, which implies he was ok with being in the open. As for why the Damage Control guy knew (and, most crucially, not Spider-Man), may be because Damage Control was notified on where Nick Fury was all those years, somehow. After all Damage Control has connections and affiliations with superhero actions in some form, so they probably were informed on that matter. Or maybe it is "common" knowledge that Nick Fury was in space for almost a year, but Spider-Man was confused with that sentence because he thought he was with him on Earth.

There is no way the world wasn't made aware of the helicarrier at the end of AoU. So it's likely that everyone knows Fury is alive by the end of that film.

Maybe he watched FFH

Fury showed up in a public battle in A:AOU. Everyone should know he’s alive. As for how he knows he’s in space… with all the craziness in the MCU, I don’t anyone is surprised by anything anymore

Boo, secret jnvasion...

@Chandlerbing5 when it comes to being in space, it suggests the Skrulls operation is a joint venture with the US gov, something that was far from certain so far.