Unfortunately we just don't have any Roman sources from anything close to that time, the oldest Latin literature we have (not including epitaphs or snatches of ritual or legal writing) are Latin adaptations of Greek plays, and they begin a century after Alexander's death. The earliest Latin history we have as a text (that is, not as a fragment quoted by a later author) is Cicero's De re publica, which has a reasonably brief outline of the development of Rome's constitution. Even if you step outside of Latin literature, the earliest really extensive description of Rome is from Polybius, a thoroughly Hellenistic figure who is not a reliable guide as to what a fourth century BCE Roman would think about Persia. Later sources draw connections between Rome and Alexander, such as Livy's famous "what if" speculation on the result of a war between the two, and Arrian describing a Roman embassy that met with Alexander in Babylon. But this was all from after the time Rome had conquered the world, so to speak, and even if we take Arrian's story as literal truth there isn't much that can be drawn from it.

We can speculate that Rome's increasing prominence at that time meant that it was not exactly a poor backwater, its treaty with Carthage from the middle fourth century show that it was engaged in the wider world and was a major power in the central Mediterranean (as indeed it had been for centuries, despite the later Romantic notion of its early days as a backwater village of shepherds), and that its maritime connections almost certainly meant news of the goings on in the eastern part of the Sea would have reached it. But it is still speculation, we just do not know what sort of things Romans of the time would be talking about as regards to places over the sea.