We'd see its effects fairly early, as a lot of black holes that would move in such a way are formed in binary systems, and are then accelerated when the second star goes supernova, this allows it to form a 'bow-shaped shockwave' of gas and dust infront of it, which can emit eletromagnetic radiation, we have actually observed these shockwaves before.
You would also see the black hole interact with the Sun's bow-shock, where the stellar wind interacts with the interstellar medium, the gas which would heat up as it approached the event horizon would produce a lot of high intensity radiation, which would be noticed by satellites in Earth orbit.
The black holes gravity could also cause an increase in the number of comets and other Trans-Neptunian Objects heading towards the inner solar system and possibly significant gravitational lensing of background stars.
However, just as interactions between star systems are unlikely, so are interactions with black holes, especially as the majority of nearby stars are small red dwarfs, which would not reach large enough masses to form black holes, so it really isn't worth worrying about.
It would be impossible to avoid such a catastrophe as you would need something of a much larger mass to deflect such an object which would be even more disastrous, but really the chances of such an event occurring are incredibly minute, so it really isn't worth worrying about.