The earliest certain attestation of the solar calendar in Egypt dates to the reign of the Fifth-dynasty Pharaoh Neferirkare, in the mid 25th century BCE. There are also possible earlier examples, but the best we can say is that the calendar was certainly in use early in Old Kingdom Egypt, and perhaps much earlier.
Current consensus credits its adoption to somewhere in the Second Dynasty, in the 28th or 29th century BCE. However, this understanding may change based on evidence from future excavations.
If you are interested in more detail, you might like to consult Marshall Clagett's 1995 book, Ancient Egyptian Science: Calendars, clocks, and astronomy (link below).
The Ancient Egyptian priests and officials would certainly have been aware of the 'missing' quarter day.
It is believed that the calendar was originally based based on the appearance of the star Sirius in the Egyptian night sky (Sirius is one of the brightest stars in the winter sky in the northern hemisphere). We have a large number of surviving records of the Ancient Egyptians recording the date of the first rise of Sirius on their calendar.
Thus, they would have realised fairly quickly that the 365 day calendar was inaccurate (after 4 years, Sirius would have appeared a day late!).
However, it is not clear that this would have mattered. The calendar seems to have been important for civil and religious purposes, but probably far less important for Egypt as a whole.
As an aside, the 365 days solar year was divided into 12 months of exactly 30 days, together with an additional 'intercalary month' of 5 days on which were celebrated the 'birthdays' or 'feast days' of the five main gods of the Memphite Cosmology: Osiris, Isis, Horus, Set, and Nephthys.
Dates were generally recorded as the Pharaoh's regnal year, followed by the month of the civil calendar, followed by the day of the month.
Although the civil calendar was a solar one, the lives of most Egyptians would have been governed by the three 'seasons', dictated by the cycles of the Nile:
- Akhet (Ꜣḫt), or 'Flood'
- Peret (Prt), or 'Emergence'
- Shemu (Šmw), or 'Low Water' / 'Harvest'
(hieroglyph images from Wikipedia)
The exact dates of these 'seasons' varied year-by-year, dictated by the vagaries of the Nile flood. Dating the rise of the Nile to a solar calendar would probably have been of little importance to the vast majority of people in Ancient Egypt.
The various Egyptian calendars are discussed at length in Ancient Egyptian Science: Calendars, clocks, and astronomy by the historian Marshall Clagett.