I'm afraid I'm not entirely sure what you mean. Alexander the Great conquered much of the Eastern Mediterranean and Rome eventually controlled most of it. Are you specifically asking how, for example, the ruling class in Seleucid or Egyptian territory reacted to the conquest of the Greek peninsula? So to something that happened beyond their own borders, but to a culturally related people? Or is your question aimed at the more general reaction of the Greek-speaking elite of the Mediterannean to the coming of Roman influence to the entire Hellenistic world?
If you meant the first option, it is important to note that there wasn't as much of a shared 'Greek' identity as you might expect between the different Hellenistic kingdoms. Sure, they all spoke Greek and shared a cultural heritage (or at least the elite did), but a significant amount of the Hellenistic age was spent fighting each other. (This is of course simplified. The Hellenistic world shared more than that, but whether they cared if Greece was conquered is another matter.)
The Roman conquest of Greece in the second century BC did not necessarily mean that the ruling classes of the other Hellenistic kingdoms were shocked or alarmed. Up to that point in 146 BC the city states of the Greek peninsula were often at war with eachother as well and the Antigonid king held very little control over them. It was not seens as a particularly powerful or signifanct region at that time.
The other Hellenistic kingdoms and their ruling classes each had their own diplomatic dynamic with Rome. The fall of the Antigonid kingdom (which ruled Greece from Macedonia) was probably of little concern to them, since the Antigonids held very little actual power at that point, with the different Greek city states operating almost completely autonomously. The ruling classes might very well have been weary of furhter Roman expansion, but they approached it from their own kindoms or city states, not from a shared feeling of Greek identity being threatened.
If you're asking about the second case (the broader reaction of the Hellenistic world to Roman influence), that answer is a bit longer. Do let me know if that is what you meant, since there are some fascinating processes of memory and identity that start to emerge during the early stages of Roman government of the Greek-speaking world. For example, some parts of the Greek elite rejected the Roman influence (be it political, cultural or economic) as barbaric and started to look back to the glory days of Athens in the fifth century BC and imitating it. Others saw it as an opportunity and embraced the Roman expansion.
Here's some literature that might interest you if you're looking for more detailed information about the dynamics between Greek and Roman culture: