Since the moon's period of rotation around its own axis, and that of its revolution around earth is (almost) same, we always see only one side of moon. This is what was, & is taught in schools. The question is: When and How did we (humans) realized that the moon is also rotating around its own axis? Was this known before the first man-made object touched moon soil?
Lunar synchronous rotation is quite evident and a trivial logical step for any human being having a model of the Earth-Moon system that involves the Moon being a spherical object revolving around the Earth. This means that the first people to have a concept of the Moon as a spherical object almost certainly were the first ones to realize that for the Moon to always show the same features it must always be pointing the same face towards the Earth, and for that it must be revolving in such a way that its rotation gets exactly counteracted.
So, yes, this was discovered way before any person landed there, way before the space program itself, and even way before telescopes and astronomical observations in the XVI century. One of the earliest mentions of an Earth-Moon model that accounts for a spherical Moon revolving around Earth comes from Anaxoras, almost 2500 years ago. But this is probably and underestimate since there is an historic-observational bias, in the sense that many other ancient thinkers might have realized the same, even before, but we lack any documentation, which is also the reason for an ethnocentric bias, since not every culture writes and creates durable pieces of information. So your answer should be 2500 years ago, but probably even further.
How further? Possibly prehistoric! In fact this concept could have arisen even without considering the Moon as a sphere at all and even without considering the Moon as a body revolving around Earth. You only need a model that assumes that there is a relative motion between the Earth and the Moon (the Moon revolving around the Earth or the other way around is not relevant really), which is not very far fetched considering that the monthly lunar cycle across the sky was common knowledge even when Neanderthals were around. With that in mind, we should expect a change in persepective on how the Moon is seen as the relative positions between the Earth and the Moon change. But that change in the orientation of the view of the Moon is not present, thus there's only one possibility the Moon is counteracting that motion by rotating itself so that it always shows the same face. This is a simple observation that can be made with the naked eye and we don't mind if the Moon is a flat disc or a sphere, or a cylinder really, because in any case we are always seen the same characteristic topographic features since the dawn of our species. So, if humans could have made a direct naked-eye observation for this to be understood it is quite probable that they did, more so considering that the study of the Moon might have been very important for their survival (marking optimal periods for night-time hunting each month), so some interest on the issue might have been in place in many prehistoric cultures. The only way this could have never been adressed is if they avoided the actual observation that made their lifes easier or if they systematically ignored for millenia any mechanical explanation or though of this fact as a symbolic representation of something else obeying some non-conventional rules in a wildly exotic model of the Universe.
But if this is not true at least I find probable that the same ideas in Anaxagoras were shared by many others before, even centuries ago. For examplewe know that Moschus of Sidon, a Phoenitian philospher, apparently shared many of the "modern" ancient greek cosmological views, as far as 3400 years ago. His school of thought is recognized by the ancient greek atomists as their inspiration and that might have roots in more ancient civilizations (in the interface between the Paleolithic and Neolithic).
Anyways, if what you are asking instead is how and when we realised the exact mechanism by which the Moon gained its synchronous rotation, then the answer is that the process was first correctly described by british astronomer George Darwin (this is Charles Darwin son by the way) in 1879. But the main idea was really present as early as Isaac Newton's work on his lunar theory. So not only the fact that the Moon has the same rotational period as its orbital period was known long ago but also the exact way as to why this happened has been scientifically understood at least for 142 years now. This is 90 years before any human landed on the Moon and 78 years before even the Sputnik exited our world at the dawn of space exploration.