Here's part of the sky in the year 1
It is part of the sky you may know well, Orion and the dogs. I've marked the current positions of Sirius, Procyon and Betelgeuse, with green markers so you can see how their positions have changed over 2000 years. It's not a lot.
The first thing that the Babylonian astronomer might notice is that there is a pole star, one that is fixed. 2000 years ago, Polaris was more than 10 degrees from the True North. Now it is less than 1 degree. It isn't immediately obvious from a casual glance at the sky, but it would be noticeable and surprising to Babylonian astronomer.
If they started looking at the position of stars at a particular time and a particular date, they would find that things are in the "wrong" position. Of course "on a particular date" assumes a calendar, and the Babylonian calendar was not the Gregorian calendar. With more measurement they would find that the point of Aries is now in Pisces. Again, its not something you would notice at a glance, but it is within the ability of a Babylonian astronomer.
Now the stars have actually moved since year 1. But it would not be casually noticeable. It would be within the ability of our Babylonian astronomer to measure, if we asked them to. The relative positions of Sirius and Betelgeuse have changed. The Babylonians could measure angles in the sky precisely enough. But we don't know if they did, and if they did we don't know if they wrote them down.
So our Babylonian would not look up and immediately say "something's wrong". But if we 1. Prepared him by telling them to take careful measurements of Polaris, Sirius and so on. and 2. Gave them some prompts on what to look for in terms of proper motion and precession, then it would be possible for them to observe some changes.