(Napkin math time.) A black hole with 1 billion solar masses would pull Earth with a force equal to the Sun from half a light year away. It does the same thing to Uranus from a distance of 9.6 light years. It would have enough influence to disturb those orbits from much further away.

So if something similar entered our galactic neighbourhood, we would notice changes in the proper motion of stars closer to it, discrepancies between the planets' predicted and actual positions, and (possibly before anything else) predictable errors in GPS timing (since this is probably the most commonly-encountered system depending on accurate knowledge of speed, time and gravity).