Time passes normally, but creatures don't age there
From the entry on the Astral Plane in the DMG:
Astral Plane, DMG 47
Creatures on the Astral Plane don't age or suffer from hunger or thirst. For this reason, humanoids that live on the Astral Plane (such as the githyanki) establish outposts on other planes, often the Material Plane, so their children can grow to maturity.
This explains some of the apparent time-stopping strangeness that visiting this plane gives you. However, there is no mention of time itself flowing differently in this plane relative to the Material Plane.
Contrast this with the Feywild, which explicitly states that time in there may pass differently from the time in the Material Plane. There is no language like this for the entry on the Astral Plane.
Time Warp, DMG 50
While time seems to pass normally in the Feywild, characters might spend a day there and realize, upon leaving the plane, that less or more time has elapsed everywhere else in the multiverse.
There is, however, similar language for the Ethereal Plane. This is further evidence that, had the designers intended for time to pass differently in some plane, they would have made a specific mention of that.
Deep Ethereal, DMG 49
Distance is meaningless, so although travelers feel as if they can move by a simple act of will, it's impossible to measure speed and hard to track the passage of time.