I guess we should clarify what "the general person" is. European, Middle Eastern, Chinese, or African? Educated or uneducated? Upper class or lower? All of these have some bearing on the answer, and most of us will only be capable of discussing some. My own work is on the European middle ages, so that's what I'm going to be discussing here. (I think that this is doubly appropriate because I suspect that this post is a response to another post about William Manchester's (contemptible) A World Lit Only by Fire.) I hope to show, briefly, that not only were people aware of days, months, and years, but that they were deeply in tune with the rhythms of time.
So the day part is pretty easy to address, and I'll start with it. For medievals, whether aristocratic or peasant or burgher, clerical or lay, Sunday was impossible to miss. Work was forbidden, plus there was obviously Mass, and so you'd be hard-pressed to find a person unaware of the day. This speaks to a point that I hope will be clear from this post: Time was governed by the sacred. What about the other days of the week? Here, again, let's look at religious services. The Divine Office, or Liturgy of the Hours( prayers like Lauds and Vespers), differed by the day, and so anyone praying the Hours would necessarily know what day of the week it was. Okay, so that's monks and priests, but what about laity? Well, in the later middle ages, we start seeing private devotional books of the Hours made for (wealthy, and usually female) laypersons. That's not to say that the Divine Office and the Mass were the only ways to know what day it was, but they should demonstrate the point, I think.
Talking about medievals not knowing the month is a little peculiar to me because the idea that we somehow know the seasons better than people who lived and died by agriculture is, I mean, it's just not tenable. Uneducated people still understood seasons. Educated people, of course, also understood seasons. There's this fascinatingly complex methodology of medieval computus, the rationale of figuring out when Easter would be based on the lunar cycle. In a world where you couldn't Google when the full moon would be, the mechanisms for working it out were really arcane and complicated and should really dissuade us from thinking that nobody knew the cycle within the year. Also, we should consider the very old chant for the moveable feasts from Epiphany - in January each year, a priest (or deacon) would literally announce to the congregation when feasts would fall during the upcoming year, what day and month. So, I mean, that presumes an awareness of what days and months are.
I suppose that years are a little harder for me to talk about, because so much medieval time is sacred and cyclical and the procession of years doesn't fit quite as easily into that. However, there was the whole millenarian thing, where a bunch of people thought that the world would end in the year 1,000, and then again in 1,033. Jay Rubenstein cites this as a major contributing factor to the pogroms, and then eventually the First Crusade. It was a really widespread belief. So I mean we can also look at chronicles and annals that say "In this year, this happened," but if you want to really see "average folk" getting worked up over what year it was, look at millenarianism.
Obviously, I'm coming at this from the perspective of medieval religious culture, because it's what I know. I hope that this has helped to prove that no, people didn't go around blithely unaware of what day or year it was. I also realize that it's fairly surface-level, and I'll be happy to go more in-depth if I wasn't clear about something. Assuming anyone even reads this.
For a bit of further reading, I would suggest
Margot Fassler, "The Liturgical Framework of Time and the Representation of History," in Representing History, 900-1300: Art, Music, History, ed. Robert Maxwe (University Park: Pennsylvania State University Press, 2010).
Hans Werner Goetz, "The Concept of Time in the Historiography of the Eleventh and Twelfth Centuries," in Medieval Concepts of the Past: Ritual, Memory, Historiography, ed. Patric Geary et al. (New York: Cambridge University Press, 2002).
For the circum-Mediterranean world: yes, people would know the day of the week and the year--but not necessarily in the way you think.
In terms of year: the BC/AD calendar, which is of course Christian, the actual system comes from Dionysius Exiguus in the early sixth century. But while it was known enough for the occasional early monastic chronicler like Bede to pick up on it, it doesn't appear regularly, really, until the high Middle Ages (11th-13th centuries). The Jewish calendar in terms of year numbering is also a high medieval creation.
But more to the point, that's still limited to use by the educated elite. Only in the Islamic world, I think, is knowledge of calendar year, in hijri of course, widespread by 1019 CE. My assumption is that Christians and Jews living in close contact with Muslims would also be generally aware of the year in AH.
That doesn't mean the Jewish and Christian residents of Christendom didn't also keep track of years. There was certainly some of it. James Palmer reminds us of the story of Abbo of Fleury, who supposedly stood up in the middle of church in Paris and started arguing with the priest whether the year 1000 would be apocalyptically significant. (To be clear: the priest had claimed it would be; Abbo was scoring points for Augustine and Augustinian orthodoxy when he wrote down the story of his disagreement in Apologeticus).
But I think in terms of how people thought, we should look elsewhere. The world of urban churches was very different from the rural churches most peasants would frequent, for starters. Among other things: a rural priest would almost certainly not be giving a sermon in which to talk about the year 1000. (Abbo is, of course, almost certainly making up the story. But it would still need to seem realistic to his own audience.)
So: as Michael Clanchy has demonstrated with court records (albeit for later centuries), witnesses figured the year based on "years since" a major event in recent local memory, or sometimes in regnal year of the current king. And why not? What are AD and AH if not "years since" a major event?
In terms of week day: the obvious point of reference for Muslims, Jews, and Christians was their respective holy day. It's important to keep in mind that attendance at, or participation in, holy day rituals was normative, not descriptive. That is to say, it was the ideal, not necessarily the practice. But in terms of something like day of the week, not everyone would have to observe the Sabbath for the knowledge to be common.
1019 is sort of the last days of what we call the "proprietary church" system, which at the local level would mean the churches most people attended were small, and established and run by lords rather than a diocese. The priest would likely be a resident of one of the villages served by the church.
As Irina Metzler showed, even in the late Middle Ages, priests are noted in court records as receiving injuries that we might call "industrial accidents" or "can I file for workers' comp for this". For 1019, even, there are still provisions requiring that priests be freemen--that is, not serfs. Local priests were more than likely blacksmiths, carpenters, farmers when not saying Mass or dispensing sacraments. They were enmeshed in their community life. In this case, that means knowledge of week day was right there next door.
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