Why Capua? Everyone is fascinated with Capua these days. Capua is a special case because it was destroyed as punishment for supporting Hannibal, then slowly reformed. It was run by a prefect sent from Rome. It's just needlessly complicated. Let's switch to Beneventum, right up the road, which was a very typical town in early Imperial Italy.
You say your person is plebeian, but you probably mean someone who is not terribly rich -- just an average person. The plebeians were actually a large class, and many were rich. A plebeian is anyone who isn't a patrician, and there were very few patricians. Anyway, let's assume your person is lower middle class, and works in ... let's say a bakery. He is not the boss baker, just a baking apprentice. I'm making your person male for the sake of simplicity -- don't crucify me.
First of all, he is a Roman citizen in the legal sense. After the Social War between the Romans and Italian allies 91-88 BCE, almost all the communities of Italy were granted Roman citizenship. By the 1st century CE, the scars of that conflict had long healed and the urban populations of the peninsula had become well-integrated. What does it mean to be a Roman citizen? Some benefits:
ius suffragriorum - the right to vote in the comitia in Rome. Our baker might not ever make it to Rome, much less vote in an assembly. Still, he has the right.
ius commercii - the right to do business and own property. Mr. Baker may own property, but probably not. Most likely, he rents a flat in an urban insula.
ius conubii - the right to marry another (female) Roman citizen. Our baker has been married since he was about 18 years old, because that's what adults did. In this way, he is the exact opposite of a modern Italian man. Our baker doesn't still live with his mother in his mid-30s, but his mother (and a gaggle of other relatives) probably live with him.
In addition to these basic rights, our guy has the right to a proper Roman trial in a proper Roman court, and the right to appeal decisions made therein. He cannot be tortured or whipped in the course of such a trial, and can elect to go into exile instead of suffer the death penalty (usually). If he gets tired of the baking life, he is qualified to join the Roman army. By the middle of the 1st century CE, the army career is actually quite beneficial and desirable: well-paid, security in retirement, and only a low-to-moderate risk of being skewered on a battlefield a thousand miles away.
These were all technical, but you are asking if he felt, in his hidden heart, that he was a real "Roman." The answer is almost certainly "yes." He probably speaks Latin primarily, though he learned a bit of Oscan from his grandmother. He is dimly aware that his city was once a proud Samnite place, of the Hirpini tribe, but he has little understanding of what that means. If he has had the time and opportunity, he may have listened to some recitals of Livy's history about the long history of his town, but probably not. When he goes to the forum, he is either going to the basilica for a trial, going to the comitia to vote or hear a speech, or just passing through. He doesn't have time to hang around and listen to literature recitals: he has bread to bake in order to support his family.
When he moves through Beneventum, his experience of the topography is wholly Roman. He bathes in the thermae, looks up at the imposing architecture of Roman temples to Jupiter, Minerva, Juno, and perhaps the divine Augustus. He shops in the cryptoporticus, the ancient equivalent of a shopping mall. He has Roman coins in his pouch (some even minted in Beneventum), and he makes the trip to the nearest amphitheater and/or circus as often as he can to see the games and the races (I don't actually know off hand what the closest is -- maybe Capua). He probably lives in a tenement flat which he rents in an insula (a large apartment block), and eats just about the same diet as the urban masses of Rome -- his quality of life might even be a bit better than his social and economic equal in the City.
When he hears that the legions have delivered a good whipping to the Germans, or have invaded Britain, he is glad, and feels that he is a apart of that -- as the glory of the SPQR is enhanced, he feels that he is participating in that. He understands that he has no influence on the goings-on of the great in the City, though he always votes in the local Beneventum elections and feels that he is connected to and has a stake in local political life. If he is literate, he has scratched graffiti on the walls of the local brothel in support of this or that candidate for duovir or aedile over the years. He positively loves the Julio-Claudians and fervently supports the Emperor. When the Imperial retinue passed through Beneventum on the way to Brundisium in the South, our guy was there on the parade route to cheer and wave. He and his buddies talked about it for weeks afterwards as they drank in the taverna.
If is is moderately well off, he owns one or two slaves -- perhaps a woman to help his wife, or a man to help him with general tasks. The slave is from some distant place like Mauritania or Germania. He doesn't beat the slave or mistreat them in general -- he trusts them as far as any Roman trusts a tricky slave, and they are generally members of the family -- just members who are not technically free. When he encounters non-Romans on the street (and this is often), he feels a sense of superiority to them. Even if they are more rich and fancy than he is, they are provincials and he is Roman.