The obvious and generally correct answer would be that the reflected image would be 2,000 years old when the astronomer looked at it; he would be looking into the past. And we'll ignore the technical issues of such a telescope and mirror.

An important factor, though, is how long the mirror has been there.

If the mirror is less than 1,000 years old, then the observer couldn't see the mirror at all, nor anything reflected in it. The light traveling from the mirror towards Earth would not have reached the Earth yet, so the presence of the mirror would be undetectable.

If the mirror is 1,000 years old or older, then the observer would see the reflected Earth of 2,000 years prior.

It might seem like you'd have to wait 2,000 years to see the Earth's reflection (1,000 years to get there, 1,000 years to come back), but whether the mirror was there or not, the photons leaving Earth are already streaming out into space. The moment the mirror is put in place, it begins reflecting that light, and the image, back towards the Earth. The image that reaches the mirror is already 1,000 years old, and is immediately turned around by the reflection, so it spends another 1,000 years traveling back towards Earth where it can be observed by the astronomer.

So by putting a mirror 1,000 light years away, you at first see nothing, having to wait for 1,000 years for the light from the mirror to reach Earth, at which point you see Earth 2,000 years previously.

It would be another 2,000 years before someone could see the original observer's reflection in the mirror (1,000 years for that image to get to the mirror, another 1,000 for the return trip). But if a telescope was pointed at the spot where the mirror WOULD be, 1,000 years before the mirror got there, then the first images you would see in the mirror would be a person looking "at" the mirror (which would be visible 1,000 years after the mirror is put in place, since it has to make the return trip). If you start looking at the spot the mirror will be at more than 1,000 years early, though, anything before 1,000 years early won't be seen because the image will have passed the mirror before it can reflect it back.

So, brief summary...

You can't see the mirror until 1,000 years after it is placed.

Once you can see the mirror, the image you see in it will be 2,000 years old.