Hi! I wrote about this a while ago, here. The TLDR, Tacitus was writing about a century after Augustus died, and that's where we get the idea that the Republic fell, but it's a problematic idea, because what a republic was meant very different things to different people, and some people (Cicero for instance) thought the Republic was falling all the time. The idea stuck in the modern conception, I argue, largely because of Hitler.
I'm happy to field further questions, if any pop up.
The short answer - it would have been pretty obvious to the intelligent observer very quickly.
Long Answer
Part 1: When Did People Realize They Were Not In A Republic
So Augustus defeated Marcus Antonius (Marc Antony) in 31BC, and at that point the military hegemony of Octavian would have been obvious, however Octavian understood from history that military supremacy in Rome did not guarantee sustained power. Further, giving one absolute power such as making one self dictator for life such as Gaius Julius Caesar puts a target on one's back.
Octavian's power was based on three pillars - auctoritas, tribunician power, and the "maius imperium." Auctoritas is the Latin word that would be a combination of authority, prestige, and respect. Auctoritas was an idea taken very seriously in Rome, and the authority commanded by Octavian was extreme. When Caesar died in 44BC, Octavian was only 19 and by 43BC was enshrined via plebeian law as one of the triumvirs in the Second Triumvirate. He was one of the most powerful people in a Republic where traditionally 10 years before he would normally have been even allowed into the Senate. The astounding amount of successes Octavian achieved all before Senatorial age would have been very impressive, and the military weight behind Octavian after the defeat of Marcus Antonius would have been terrifying. All combining to give him a large amount of auctoritas.
The second pillar was Tribunicia potestas (tribunician power) and constitutional tweaking. Rather than declare himself as having ultimate authority in the manner of Julius Caesar, Octavian instead gave himself piece by piece crucial positions that already existed within the traditional concepts of the Roman constitution. Foremost of these, he gave himself "tribunician power" that is the power of the tribune of the plebeians. The Tribune was a crucial role, holding a veto on legislation, sacrosanct physical inviolability (you can't beat up the tribune or kill him, however that didn't stop people in the past with tribunes who pushed the system too far such as Tiberias Gracchus being murdered), and the claim of being able to represent "the people." This was so crucial a power to Octavian, that moving forward emperors' reigns were marked as having started from when they "assumed tribunician power. There were additional constitutional changes, such as creating a load of new administrative positions called the ordo equitum (knights' order would be a loose translation). These were a series of key positions that Octavian was allowed to appoint. Elections would still fulfill the traditional cursus honorum (e.g. consul, aedile, praetor etc) from the senatorial class, thus Octavian could claim he was not changing the constitution or taking on the roles of these positions. However a) he massively undermined these positions by creating new, arguably more important positions such as the city prefect, imperial governors, grain officers, and b) he made sure that people he wanted would get elected to these traditional people. But crucially there was at least a facade of traditional constitutionality. Further, Octavian raised many new people into the Senate, giving him the upper hand in elections and a large group of senators loyal and in debt to him for their position. Finally whenever Octavian wanted something, he made sure to have the Senate vote on it for him. In the Res Gestae Divi Augustus (Life and Deeds of the Divine Augustus), Octavian is described as being given power by the Senate rather than seizing power for himself. Even the name "Augustus" was a title bestowed, officially, by the Senate on Octavian, though realistically they would give him nearly whatever title he wanted (so long as it wasn't Romulus).
I can now call Octavian Augustus
Third pillar - maius imperium - the supreme military command of the Empire. This one is obvious, Augustus had the love and adoration of the legions and this gave him a lot of power. Importantly, Augustus made sure that the legions would remain under his control when he was not campaigning. Canny politician, he did this in a way that would again placate the Senate. Once Marcus Antonius was defeated, Augustus "returned" Senatorial control over many provinces in the Republic. He made this out to be a gracious act, restoring traditional Republican control. However Augustus said that he needed to maintain control over the border provinces and certain provinces that had been part of the rebellion of Brutus and Antonius/Cleopatra's rebellion. What was important about all those provinces? Border provinces and provinces recently pacified would be the ones in which almost all of the Republics legions would be operating. Thus, Augustus got the PR victory of restoring control of provinces to the Senate that had for years been effectively out of their control, but at the same time ensuring that he had total military hegemony through legal means.
So now we know how Augustus ruled, we can see that it was extremely obvious that the Republic had changed, however people still referred to it as a Republic, and it functioned in familiar ways.
Part 2: Would it be different for patricians vs equities vs plebeians?
This is very broad, but I'll try break it down. The Senators and patricians would most definitely have been aware. As would the equites/publicans (that is money lenders, bankers, businessmen). Plebeians is a very general term though. Plebs could be everything from a wealthy banker owning large tracts of lands, to a mud shoveller at a brick maker's in a small town. In this regard I am going to defer to Jo-Ann Shelton from "As the Romans Did" -
The opposition to the principate (Augustus' imperial rule) came mainly from the senatorail class in Italy, and it was effectively silenced by Augustus (as I the reddit poster discussed above). We have virtually no information about how the lower classes in Italy and in the provinces felt about the changes made by Augustus but it is unlikely that many people felt anger or expressed opposition. The lower classes, particularly outside of Rome, probably felt far removed from what must have seemed to them a power struggle between the aristocratic senators and the aristocratic Augustus. Moreover, Augustus' rise to power had brought peace and stability to Roman society.
So yes, according to Shelton at least, it would matter depending on your economic status and educational background. In terms of dissent and its consequences, I am too tired and hungover to elaborate more on that right now, but shortly put - Augustus had fought years and years of wars to silence his opponents. He was not one to tolerate political enemies.
Sources: Frank Abbott Society and Politics in Ancient Rome
Dio Cassius Roman History
Jo-Ann Shelton As the Romans Did
Tacitus Annals
Lecture Notes - University of Toronto.
Loose Reference to Res Gestae Divi Augustus.
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