For a particular star and location, the start of a Great Year (being the length of time for one precessional cycle to complete, roughly 25920 years) can be taken to be the year in which the star has its lowest altitude at culmination.
For the next 12,960 years the star's altitude at culmination increases until a maximum is reached. After that, the altitude decreases until it returns to its minimum again.
Choosing Betegeuse (α Orionis) as the star, and my back garden as the location, to a first approximation the midpoint of the current Great Year will therefore occur in 2080 AD.
At 00:04:47 (UT) on winter solstice 2080 Betegeuse will reach an altitude of 45° 34.374' at culmination (winter solstice is chosen because the Sun is diametrically opposite Orion in the sky at that time of year).
12,960 years ago, at the winter solstice (where then the Sun was directly above Orion in the sky, being 'half a cycle' back from where we are today), Betelgeuse did not rise at this location - it grazed below the southern horizon by roughly 2° at culmination.
This is because the diameter of the precessional circle traced by the north celestial pole is 2 * 23.5° (approx) = 47° which is greater than Betelgeuse's maximum culmination altitude.
So I am going to date the start of the current precessional cycle to 10880 BC (give or take).
At that time, the vernal equinox point was in the constellation Virgo and Vega was the "bright star of evening" closest to the north celestial pole.