Time zones are a 19th-century invention that are only necessary if you are trying to synchronize clocks over long distances. They were invented because the speed of railroads made that a pretty problems, both because of schedules and because they shared track. Time zones are not "real" in the sense that most people mean by "real": they are an organizational principle designed to make human activity a little easier to synchronize, not some inherent property of time itself.

If you measure "noon" as the Sun being at its highest point (solar noon), then that will vary predictably by your longitude. If you are not traveling very fast (e.g., under foot or animal power or sail) you will not notice any practical difference as you move from one place to another. You will not notice the Sun setting and rising at different times at all; the differences you get by moving that slowly are too minor to notice.

This does not mean that people didn't know that time worked this way. The learned among them did, because this is a simple extrapolation from knowledge of the fact that the Earth is round, which was known since antiquity. But it had very little practical relevance in the medieval period, except in a few circumstances: people who did astronomy, or people who used astronomy in some practical way, like navigators or, interestingly, travelers from the Middle East who needed to calculate their position so they could pray facing the right direction and at the right times (using tools like astrolabes, which were sort of astronomical calculators). Even this is a little removed from the time zones question, but I want to just impress upon you where this kind of knowledge of the heavens would come up in a practical way, because it is not where it would come up later.

In the Age of Exploration, the calculation of time became very important because comparing what time it is at a given place (e.g., what time it is in London) to what time it is where you are locally (e.g., when the Sun is at "noon") lets you calculate your position. And once you can sail vast distances, you can run into things like a missing or gained day as you circumnavigate the Earth, as Magellan's crew discovered to their surprise. So this is where you start to see the first attempts to synchronize time, which get you into the minute differences between different "local" times.