Of course it depends on the size of the hole. The only way you are likely to be able to spot a small black hole is by its gravitational effects. If it is a Sun mass hole there would be a lensing effect. The space surrounding the Solar system is empty as far as we know. If it travels at 100 km/s then it is at a distance of 1/3000 lightyear initially.
Would we be able to see deformation of the images of the stars?
Yes.
Microlensing effects can be seen. We could computerize the observations and let them give a warning if it is detected. Would we be able to tell its distance? That is difficult, but assuming it to be one year away from here we can estimate its mass.
If the warning is given then what? Nothing. We would live in fear (that is, the people who know) without being able to do anything about the horrific consequences (the solar system will be disrupted). So it is maybe better not to know. The days that the disrupting starts will tell. If the black hole mass is small (but big enough not to evaporate substantially in a year) it remains to be seen if it can be detected at all. You would only see a black sky with stars. But the impct on Earth will be visible for certain. Black holes are, in that respect, far more sneaky than asteroids, which you can see. I am not sure how big the mass must be for the BH not to vaporize substantially though. Maybe others can tell. It is not so difficult to calculate.
