One basic problem with this is that we really have no real idea of what a Roman can know without looking up, because by necessity we only have what was written down and carefully composed, not what was thought up on the fly. Furthermore, nobody thought to write a book of common knowledge. Another problem is that the, shall we say, historiographer tendency among Romans, particularly Late Romans, wasn't particularly strong. Certainly there were historians who could produce a stunning range of material (Ammianus Marcellinus comes to mind) but if you go through the works of an intellectual like Augustine you see citations and quotations of poetry, philosophy, satire, rhetoric and more pure literature but not as much of history (Livy would be a major exception, but even he is most quoted on his more "literary" first ten books). It is very common to run across references to Cicero, very rare to see references to Tacitus.
That being said, by giving 50 CE as the date you make things a bit easier for me, as this would be smack dab within the Julio-Claudian Dynasty (in fact, in the middle of Claudius' reign). The Julio-Claudians are by far the most famous collection of emperors, to the point where Caligula and Nero are recognizable names among the general public in ways that Domitian or Septimius Severus are not. This was also in a very real way true in ancient times, which can be neatly illustrated by seeing just how much material survives on them. Much less survives for later periods--tellingly examples are that Suetonius who dealt with the Julio Claudians and Flavians gives much abbreviated accounts of the latter, and of Dio Cassius' histories nothing survives after the Julio Claudians. because for a work to survive it needs to have had a baseline of popularity, this is a pretty good indication that works dealing with the Julio-Claudian Dynasty were more popular than those dealing with later ones, and thus knowledge of this period was greater.
So in general it is likely that your hypothetical educated Roman would know who was emperor in 50 CE because that was a fairly well known and popular period. A comparison in the English speaking world would be something like the Tudor dynasty. It is less certain whether he would have deep knowledge of the Flavians, Antonines or Severans.
To head off the request for sources, my knowledge of Late Antique intellectual history is primarily from Peter Brown, Augustine and Late Antiquity, my knowledge of Latin literature's survival and the relative popularity of works comes from Conte Latin Literature.
Emperors and famous figures were remembered not only due to histories about them which may or may not have been read as 'general knowledge', but due to their monuments and statues, from which people learned their names and exploits. Emperor Claudius was an able adminstrator who built many acqueducts and public works that lasted long after he was dead. One of his aquaducts was named after him, the Aqua Claudia, which was a prominent landmark in Rome, so this may well have led to local Romans remembering his name, and even when he was Emperor. It was still in use in 300 AD, although it had had to be extensively and repeatedly repaired throughout the 2nd and 3rd centuries.
It is known that his name was also added to the Triumphal Arch in Pavia (of which nothing now survives), which still stood in 300 AD. However this was only a name on the edge of the arch, not one of the ten statues of members of the Judio-Claudian family that crowned it. And the name did not indicate that this man later became Emperor.
A Temple was dedicated to Claudius in Colchester (Camulodunum) in Britain. It is likely that a Roman living in Colchester would know who Claudius was because of this. The Temple of Claudius was the largest Temple in Britain at the time, and underwent expansion in the 4th century, so it, and the name of Emperor Claudius, would have been well known, perhaps in other parts of Britain as well as Colchester.
However, despite the presence of these monuments, and important edicts that would have lasted as the foundational law and custom for centuries, it is impossible to know what the 'common knowledge' was, and what would have only been known by someone who had studied it, or been particularly inquisitive, or had read a copy of Suetonius' "Lives of the Twelve Ceasars", which was very popular when it was written in the 2nd century, but I don't know whether it would have been general reading for every literate Roman in 300 AD, or merely those of a scholarly persuasion.
The Roman monetary system plays an important role in this. The Augustuan monetary values for roman coinage were introduced by Augustus in 27 BC and where only abolished by Dicoletians reforms in 301 AD. This means that a coin minted at any point during this period was legal tender until 301 AD. There are hoard finds that indicate that coins of the early emperors (such as Claudius) were in circulation throughout this period. This is signficiant because the coins featured the emperors. The legend on the coin would be able to tell any literature Roman, which emperor they were looking at. The craftmanship of these earlier coins was of a very high standard and as a result many of the emperor's busts would be distinguishable and recognisable just by their profile. It doesn't tell the citizen what the exact time period of the emporer's reign was but they would be aware of the emperor's featured on theses coins. Old coins would show up to 300 years of wear so that would give additional indication as to the age of the coin.
I think another good question is: would they actually be able to look it up? Did the average Roman have access to a library or some sort of expert historian? I guess for this we would have to assume they were living in some sort of heavily populated area.
I would say people would be familiar with the Emperor at the time simply because any upper class roman is likely paid in gold or silver coins with the name of the current Emperor stamped on the face. Assuming they look at their coins every one in a while, they know who the current emperor is.
For example, this coin of Septimius Severus very clearly has his name written on it and considering that any upper class roman would be paid with it, everyone knows who the ruler at the time is.
http://ettuantiquities.com/Ebay-Pictures/Emperors-new-large.jpg
Any emperor who lived any decently long time minted coins with their portrait on it. Asking any upper class roman citizen at any time, "who is the emperor?" is like asking and englishman, "who is on British 1 Pound note?"
Source: Numismatic records of ancient Roman coins.
