What is the closest distance for an undiscovered black hole? How close can a stellar black hole, an inactive old one, get to our solar system before we notice it? And then, how long to get to the inner solar system at typical speeds?
I know it is very unlikely and that is not what the question is about. I have already worked out the probability of a black hole getting close to the solar system and it is tiny. But in that unlikely event that one was headed straight for us, at what point would it be noticed?
I am looking for an approximate distance in au's or light years or similar. And a time in years for it to get here.
If it passed in front of a distant star presumably it would show an Einstein ring? If it just distorts the star field it might not be that noticeable as remember by hypothesis it is coming directly towards  us. It would have a gravitational effect on the many dwarf planets we are tracking beyond Neptune, and we'd notice they don't follow the predicted orbit, also on our spacecraft, but by hypothesis it comes from any arbitrary direction. And if it had an accretion disk then we'd see it, but if old and inactive, how much of an accretion disk would it get from traveling through our Oort cloud?
Those are a few points to consider in your answer. It is not an easy one to answer. I've tried in rough calculations myself. Thanks for your help!
I've been asked to explain why this is different from my previous question about a neutron star. There the answer was that a neutron star can be seen about a third of a light year away by its intrinsic radiation even if it is very old. 
I can't use the previous answer to get a minimum distance for a black hole, because a black hole produces no light, nor does it reflect light and so, unlike a neutron star, it can't be seen directly. It is also going to be heavier - a neutron star may be as light as 1.1 solar masses and a black hole most likely at least 5 solar masses. Gravitational effects are likely to be an important part of any answer, and perhaps the ability to form an accretion disk from the Oort cloud. It is just a different problem.
 A: You say it is not a duplicate, and I note you did not accept my answer to that question, but the answer here is similar in nature if not in numerical value.
There are all sorts of things to consider here and I doubt there can be a definitive answer.
First: how many black holes are there - or more pertinently, what is their density in the solar neighbourhood.
There are about 1000 stars within 15pc of the Sun down to about $0.2M_{\odot}$. Most of these are main sequence stars that are less massive (and more long-lived) than the Sun, with odd exceptions like Sirius and Arcturus. About 10% are white dwarfs that have evolved from objects with initial masses of $1-8M_{\odot}$. If we assume there are 900 stars within 15pc that were born with $M\leq 1M_{\odot}$, then we can integrate an assumed initial mass  function and further assume that all stars with $25<M/M_{\odot}<130$ have already ended their lives as black holes. The upper limit is uncertain and likely determine dby the initial masses of stars that end their lives in "pair instability supernovae" that leave nothing behind. Fortunately, the steepness of the mass function ($N(M) \propto M^{-2.3}$) means the upper limit does not affect the numbers too much.
The lower limit is also uncertain, could depend on things like initial rotation and chemical composition, and this could change the [small] numbers of black holes by factors of a few.
Thus the fraction of all stars that end up as white dwarfs would be
$$f_{\rm WD} \sim \frac{\int^{8}_{1} M^{-2.3}\ dM}{\int^{130}_{0.2} M^{-2.3}\ dM} = 0.115$$. This is in reasonable agreement with observation, but will be slightly overestimated because not all stars more massive than $1M_{\odot}$ have died.
Armed with some confidence that this calculation works for white dwarfs, we can do the same calculation for black holes.
$$f_{\rm NS} \sim \frac{\int^{130}_{25} M^{-2.3}\ dM}{\int^{130}_{0.2} M^{-2.3}\ dM} = 0.0017.$$
i.e. If there are 1000 total stars within 15pc, there should be 1-2 black holes. This is likely to be an overestimate because a large fraction of black holes are created in supernovae and obtain a large momentum kick that could give them velocities of hundreds of km/s (the evidence for this is seen in the high velocities of many, admittedly lower mass, neutron stars). That means they should be under-represented in the Galactic disc and some will have been ejected from the Galaxy. This is probably a factor of $\sim 2$ effect.
This means that the nearest black hole is probably within 15 pc ($\sim$ 50 light years).
Second: Could we actually see these nearby black holes? The answer is no, unless they were in close proximity (i.e. in a binary system) with another star. Travelling at let's say 30 km/s (a typical relative velocity for stars in the local galactic disk) the black hole would spend $\sim 16$ years covering the last $\sim 100$ au into the solar system. I don't think there is any doubt that the influence of a $\geq 5M_{\odot}$ object within 100 au would be noticed, so there should be plenty of "dynamical" warning from precise observations of solar system objects. Whilst black holes could be travelling faster, they would be less likely to be in the Galactic disc. 
Third: I should point you to another possibility that I addressed (and dismissed) in https://astronomy.stackexchange.com/questions/16578/will-gaia-detect-inactive-neutron-stars/16699#16699 That is that Gaia might see the gravitational lensing of background stars by a foreground and nearby ($\leq 1$\,pc) neutron star. In the answer referred to, I showed that this is possible but rather unlikely ($\sim 10^{-5}$). This probability increases as the square of mass, so may be two orders of magnitude higher for black holes.
In conclusion there is little reassurance to be given other than that I think we would get at least decades of warning based on accurate metrology of solar-system objects. The likelihood of a black hole disrupting the solar system is very low and since they are many orders of magnitude less common than "ordinary stars" (which would do just as much damage!) and none of these ordinary stars show much likelihood of coming nearer than 10,000 au to us in the the forseeable millions of years (e.g. Bailer Jones 2018) it would be unfortunate in the extreme to be struck by something that is orders of magnitude rarer anytime soon. 
