So it should first be noted that universities were not centers of research until the 19th century. So if your question is rooted in the assumption that they were where research was done in, say, the medieval through early modern periods, that is largely not the case.

Generally speaking, and this is a broad generalization indeed, prior to the modern period you see pretty stark separations between the worlds of what we might call "learned knowledge" (which would include theoretical knowledge about the natural world, like cosmology and astronomy) and "craft knowledge" (which would include things like "how to make X" where X could be pretty much anything difficult to make). These worlds were not always totally divided, and bridges were sometimes made between them by specific individuals and in specific areas of work, but it is a useful "template" going into it to think of these as different social environments with different people participating in them for different reasons and through different institutions of education.

For professions requiring formal knowledge, like knowledge of writing or law or mathematics or even astronomy, we have evidence from ancient times of what we would essentially call schools today. Some of these are quite famous, like Plato's Academy and Aristotle's Lyceum, but most are not. These kinds of places exited to pass on the kind of knowledge you are talking about, and they did have written media through which to do it (just not the printing press). The exact types of "schools" is going to differ depending on when and where you are talking about in the ancient world; whether these are "precursors to universities" is going to depend on what you mean by "university" (most people take "university" to just mean "educational institution" but historically it is better to think of them as "independent union of faculty and students"; neither of those definitions, though, actually have anything to do with research by themselves).

For craft knowledge, you tend to see apprenticeship systems and guilds. The goal here is to transmit tacit knowledge — the kind of knowledge you learn "in your hands" through long experience — and that is still how we teach many crafts today (even in formal knowledge institutions).

Could a single individual have developed both the theoretical knowledge (astronomy) and craft knowledge (gear-making) to make the Antikythera mechanism? It would have had to have been a rather exceptional polymath, but we have examples of such people from later periods (Robert Hooke comes to mind as someone who had the relevant knowledge and skills to make something like that, albeit over a thousand years later), so it is not impossible, but I agree that it seems more likely on the face of it that it was some kind of collaborative effort. But who knows.

Lastly, I would note that attention to "physical experimentation" here is ahistorical; this was not at all common in the ancient world, and didn't really become common until much later (unless one counts things like alchemy as physical experimentation). Most of the "sciences" you would learn in the ancient world were some mixture of deductive and observational. The use of "experiment" as a means of advancing knowledge was very uncommon until much later (experiments are about creating artificial conditions in order to isolate variables; much of ancient philosophy and epistemology is about finding the "natural order" of the world, and would have seen experiments as not necessarily indicating that).