The sun is in the milky way galaxy, funfact, all galaxies are milky ways, and along with the sun, so is every star we see in the sky.
So while the sun has moved quite far in the past thousand years, so did all other stars, at most there would be a 5-15 degree change, but overall it would not be noticed, even if you picked up a roman astronomer and let em see todays night sky.
As others mentioned the movement of stars around the galaxy just takes such a long time we only see it in the timescale of millions of years.
There are other effects that change the night sky.
Axial precession can be seen in the order of 100 years, with careful measurement.
Not so much the shape of the constellation but where the constellations are relative to the north, and which star is the north star also changes
https://en.m.wikipedia.org/wiki/Pole_star#Precession_of_the_equinoxes
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