About the minimum distance, well, it can be also as near as alpha centauri.
If they are a thousand years more advanced than us, starting from the idea that they used the same communication medium (radio wave) we use and that they used it at more or less our development level, their waves reached us about 1000 years ago, when we were not able to receive them (well, any time before the last 250 years). Now they can simply use some transmission method we are simply not able to intercept or even understand.
While saying "a millennium ahead of humanity, technologically" don't really say anything on their technological level, I suppose that they are more than able to mount a big engine (with some smaller ones for trajectory correction) on a small asteroid and launch it against us. As they are on the verge of FTL, a not so out of reach supposition is that they can aim well at the long distance or their FTL will end badly.
If they are able to apply to the asteroid a small acceleration (let's say just 0.5 g) for a year you end with about 5% of c (a little less of 15800 Km/s). Keep doing this for some more years and you end with a pretty decent fraction of c.
At this point, you can simply turn off the main engine and let the asteroid go, eventually allowing for some course correction every now and then.
It is not super sneaky (we can find out it, eventually) but its biggest advantage is that we can find it only if we know where and when to look. Even if the engine is on, we must look precisely at it to be able to detect it. The demonstration are all the asteroids we found just days before they fly-by the earth or even after this.
But in the end it don't really matter if we can see it, we have no way to stop it.