The movie is set in 2259, McCoy says the crew member in the cryo-tube is 300 years old.
One thing you should remember is that many people (especially those who aren't Vulcan or androids) aren't very precise.
Perhaps McCoy would have said that the time span was 300 years if it was anywhere from 275 to 325 years. Perhaps McCoy would have said that the time span was 300 years if it was anywhere from 250 to 350 years. Perhaps McCoy would have said that the time span was 300 years if it was anywhere from 200 to 400 years.
Since we don't know it is safest to assume that the crew member in the cryo-tube was somewhere between 200 and 400 years old. And that he was born about 20 to 60 years before he was frozen, thus making him frozen about 140 to 380 years earlier.
In "Space Seed" Khan's people were frozen and left Earth in 1996 two hundred years earlier, which should be somewhere between 100 and 300 years. In
Star Trek II: The Wrath of Khan fifteen years after "Space Seed" Khan left Earth in 1996 two hundred years earlier, or about 100 to 300 years. Since "Space Seed" was fifteen years before
Star trek II: The Wrath of Khan "Space Seed" must be 100 to 285 years after Khan left Earth in 1996, and thus about 2096 to 2281, and
Star Trek II: The Wrath of Khan must happen 115 to 300 years after Khan left Earth in 1996, or about 2111 to 2296.
Note that those dates are given in the calendar used in "Space Seed".
Star Trek characters rarely state which calendar they are using when giving dates, thus it is illogical to assume that they always use the same calendar, let alone your calendar.
There are many examples of governments changing their official calendars. Some old people still alive today in Iran and Thailand can remember joking about suddenly becoming thousands of years older or younger when the official calendars were changed. And naturally the United Earth government would sometimes change the official Earth calendar for symbolic political reasons.
In the first one or two seasons of
Star Trek: The Next Generation the characters seemed to use an Earth calendar where Data was a member of the Starfleet Academy class of '78, and Earth already had rapid interstellar travel by the late 20th century of that calendar - perhaps using the calendar from "Where no Man Has Gone Before". But in later seasons the characters began using a calendar in which the first season was in year 2364, the second in 2365, the third in 2366, and so on. This was the calendar used by the "corpsicles" or revived frozen dead in "The Neutral zone". I call it the "Corpsicles' Calendar, abbreviated CC. So it is logical to assume that they became famous after returning to Earth and the United Earth government adopted the calendar used in their era as the new official calendar of Earth.
But the creators of
Star Trek: The Next Generation never noticed that they were replacing one Earth calendar with another onscreen for anyone to observe, and assumed that all Earth dates in every version of
Star Trek were given in one single calendar, our current Gregorian calendar. So they never bothered to figure out which dates were given in which Earth calendar when writing the
Star Trek Chronology.
If the date of 2063 in
Star trek: First Contact is in the CC calendar, then since the Third World War in about 2053 CC must be identical with the Eugenics Wars in about 1996 SS (""Space Seed" calendar), the SS calendar must have a year one that is about 57 years after the year one on the CC calendar. And there is no proof that either year one is the year AD one in the Gregorian calendar.
IMHO every single official future Earth date in
Star Trek is highly likely to be inaccurate, because I don't believe that writers have the authority to arbitrarily say that their stories happen in any date that they want, even when the date they choose is impossible because it contradicts the data in their stories. I say that writers have the duty to make certain that the official dates of their stories are consistent with the dating evidence they give in their stories, instead of arbitrarily decreeing they are a round number of centuries in the future from when the stories were written, for example.
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