Would a person have time to put a shield up if they heard an arrow fired at them?
I’ll be honest, I’d be impressed if they heard the arrow being released. Bows aren’t completely silent, but as ranged weapons go, they’re close.
I’d be inclined to say, if they could hear the bow, they’re close enough that they wouldn’t have time to react. If they were far enough away to react, they wouldn’t be able to hear it.
Media tends to do two things with bows that might mess with you a bit. They give them sound effects because when you’re watching something on screen and there’s no audio, it “feels,” off, or like there’s been an error. Second, they tend to slow the projectiles way down, because it can very easily look like a jump cut, so they’ll aim for something that looks way goofier with the goal of creating a more “realistic,” scene. Neither of these practices are universal, and you will occasionally find good bow work on film, but it’s a rarity.
Blocking arrows with a shield is a real thing, but it’s more about knowing the arrows will be coming, and having your shield up in advance, rather than reacting to a surprise attack.
Someone who’s wired up to the point where if they think they hear a bow being released, will immediately bring their shield up would be a nervous wreck, and could probably be startled off the ramparts by an unexpected kitten. Just, food for thought, though that might seem like a less plausible assassination technique.