This has been asked many times on here, but the short answers are:

• The fact that "solar noon" varies with your location on the Earth is a straightforward conclusion, as is the fact that if you are on a sphere, the other side would be in darkness. Hipparchus (~190BCE) used the differences in "local time" as a means of calculating longitude, which relies on this understanding. You do need some way of keeping track of the time (so some kind of "clock," though this can be as simple as a sundial) for this (you do not need a magnetic compass; really, all you need to do this kind of astronomy is a gnomon — a stick that casts shadows).

• The first time that anyone seems to have had a real practical effect of this was during the first circumnavigation of the world by Magellan's crew in 1522, when they were surprised to find that they had indeed lost a day from their count (they tracked the days very carefully in the ship's log, and were particularly attentive to religious holidays). Again, the fact that this would be true is a straightforward result of knowing that you live on a sphere, but it is highly unintuitive (and indeed, it still feels highly unintuitive to us today, as anyone who has "lost" or "gained" a day through international flight has observed).

• "Time zones" are totally artificial and were developed in the 19th century, initially to coordinate railway schedules. All they do is make the variation in solar time "chunky" rather than continuous, allowing you to standardize time in a way that is not strictly based on the Sun's location. They frequently vary considerably from solar time.