The answer to this question certainly depends on the society and also very likely on the status of a member of society. I study Mesopotamia and calendars were developed very early on and become administratively more standardized with the expanding needs of urban society in the late 3rd millennium. While you are likely to some degree correct that calendars may have been less useful prior to the development of agriculture, remember that agriculture itself was nearly 5,000 years old already by the time we find our first written evidence. So even if it were true that calendars did not appear until the development of agriculture it is impossible to really know this due to the fact that written evidence from the periods before the development of agriculture are simply nonexistent.

Outside of agriculture and administrative uses calendars can still be quite useful. For a pastoral herder keeping track of not simply the seasons but likely even the days would have been very important. They needed to move their herds to different grazing areas throughout the seasons and moving a large herd of animals is not at all an easy task to begin with. If you move your herd from the current grazing area too late especially, you risk losing parts of your herd by not reaching the grazing area in time. You would not want to be still moving your herd as the weather worsens, so it would likely have been helpful to know when it is worth moving the herd while the weather is good and not simply doing so when the weather is clearly bad.

When discussing calendars, you have to remember that creating a calendar is in ways as simple as looking up at night and noting the phases of the moon and positions of the stars. Various stars were already used as navigational tools as well; the pastoralists would follow the visibility of specific stars to know if their path was correct. If the star is ever not visible in the night then you have an easy indication that you are moving in the wrong direction. A number of stars that were particularly bright became important calendrical tools, with the most important generally being the star Sirius. If you have never known where the phrase "dog days of summer" comes from, it refers to the recognized connection between the first visibility of the dog star Sirius and the hottest days of the summer months.

Another important aspect of calendars that would have been significant for nomads would be their use in organizing meetings. There were likely days where the pastoralists would come into towns to trade for products they could not get from herding sheep or goats. These markets would also have agriculturalists bringing their product and more importantly traders bringing goods from areas further away who would not stay in the market for particularly long. If you were to organize such a meeting in ancient times, the stars would realistically be the easiest way to do that. Even beyond the creation of more standardized administrative calendars timekeeping with the stars was likely still the most convenient way for the majority of people to keep track of the calendar.

Calendars were obviously immediately necessary for administrative and ritual purposes as urban society developed further. You needed to know when to collect the yearly taxes, when to perform certain festivals to the gods, when to go out campaigning in order to raid and loot nearby areas, among many other uses. The Mesopotamian calendar was a luni-solar calendar and it used the phases of the moon as its month reference but also attempted to keep the year in line with the seasons by occasionally adding an extra month to years. The lunar year, or 12 full cycles of the moon, was about 354 days, so 10 days less than the 364 day solar year. Following strictly the lunar year would result in the months moving slowly in relation to seasons, which was very bad considering so many festivals happened in relation to seasonal activities. This is also why you hear Muslims saying that a summer Ramadan can be deadly while a winter Ramadan is much easier. The Muslim calendar is a strict lunar calendar.

As to your question of birthdays, this was something that very likely did not occur until another important shift in thought by the Hellenistic period. Up to the Achaemenid Empire at least, astrology and astronomy were for the king most prominently and the kingdom itself. The Babylonian astrological omen series Enuma Anu Enlil, which consists of over 70 tablets of "if... then" statements relating to the positioning of the stars, is basically entirely concerned with state matters. For example, an eclipse would foretell the death of a ruler in one of four places depending on which part was eclipsed. There are numerous references to floods, crop success or failure, and other matters relating to the kingdom as a whole. These were generally unusable by the common individual, and at least for Mesopotamia during the late 2nd and early 1st millenniums we have little evidence that there was a common astrology. While we read of wealthier individuals consulting the stars, for the majority of society this was likely out of their reach.

During the Achaemenid and Hellenistic period this picture seems to have changed drastically to the point where we have found cuneiform horoscopes written for individuals. The lack of a native Babylonian king likely made the cuneiform astrological omens less important. While certainly some Persian and Greek kings paid heed to the predictions of Babylonian astrologers or at least feigned interest in them, Babylonian astrology would have found significant competition from Persian and Greek traditions building off the Babylonians. I would expect that a combination of changes from interacting with other traditions and also the rapidly deteriorating power base (and subsequently income stream) for astrologers in the court pushed them to ply their trade to more common individuals.

So to answer your question directly, this is the point where more and more people would have been interested in the exact day of their birth. Astrology uses the position of the planets in the zodiac on your day of birth most prominently, and so if you wanted to get a horoscope that was accurate you needed to know your exact birthday. I would imagine there were a lot of people who would make up a day and not be certain, but often the way you would even know your birthday was that your parents would have your horoscope cast shortly after you were born. Prior to the "democratization" in a way of astrology, I expect you are correct that very few if any individuals felt the need to keep track of their specific birthday, and would be content only to know the passage of months and years.

If you would like to read more about the creation of calendars and the early Mesopotamian calendars, John Steele has written extensively on the subject. He edited a volume called Calendars and Years: Astronomy and Time in the Ancient Near East that is recent and would provide a good list of references to look further. He also wrote the chapter "Making Sense of Time: Observational and Theoretical Calendars" in The Oxford Handbook of Cuneiform Culture. Francesca Rochberg has written extensively on late Babylonian and Hellenistic astrology, and her book Babylonian Horoscopes is the best treatment of our evidence for them.