Well, the time is obviously noon by definition(*), and the date would have been well known as it's entirely predictable. Syene is about a degree or so north of

Well, the time is obviously noon by definition(*), and the date would have been well known as it's entirely predictable. Syene is about a degree or so north of the Tropic of Cancer, so essentially the only date the sun is in the zenith is the summer solstice, and that’s the day they should measure shadow length in Alexandria. Had it been farther to the south, there would have been two such days, still well known and understood by the people of the time, and had been for millennia, and it wouldn’t matter which of the days you’d pick. (*) This was 2000 years before time zones became a thing and local solar times were disengaged from the wallclock time. Not that ancient Greeks had wallclocks.