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Assuming we are unable to see the black hole via any way such as gravitational lensing, accretion disc or whatever. This is hand wave approach just to make my setting work, due to this answer which sets it too far away.
It's approaching us with speed of 3000 km/s or around 1% of speed of light.
At what distance we would feel the gravitational effects of a supermassive black hole? Like nearby stars changing orbits, interstellar gas changing shape etc.
OK, I believe your question is when the purely Newtonian gravitational effects of an approaching supermassive black hole (SMBH) would be apparent -- ignoring the massive distortion of the star field near it, ignoring the radiation from the accretion disk, and ignoring the fireworks as Oort cloud planetisimals fall into it, etc. Ignore all the fancy General Relativity effects, too (though they won't be all that apparent until you get very near the SMBH, anyway.)
You didn't specify its mass, so I'm going to assume it's 10 million suns. (We have evidence for SMBHs as large as a few billion suns' mass, but the only SMBH we know of nearby is much smaller.) The answers won't be greatly different for larger or smaller SMBHs.
The first effect to look at is the acceleration towards the SMBH due to its gravity. The acceleration scales linearly with mass and inversely with the square of the distance, so the SMBH has the same effect as the Sun does at about 3000 times the distance. This is larger compared with the solar system, but not all that big otherwise -- orbital velocity around the SMBH is the same as the Earth's around the sub at about 3000 AU which is about 0.05 light year.
Since the SMBH is moving through space at .01c, it will move a light year/century or 0.05 ly in a couple weeks. The SMBH's gravitational impact on bodies will depend on the distance to the body, but the time during which it is within .05 ly of any object will be only about 5 years. So the region of space over which it will have a significant effect on motions is a cylinder along its path maybe 0.1 to 0.3 ly in radius. Unless the alignment is just right, there will be no bright stars affected, though some red dwarfs might be.
That means that we probably wouldn't have noticed its effects on stars' proper motions until 10-30 years ago when instrumentation became good enough to measure positions very accurately, and to monitor lots of nearby red dwarfs. (Now, of course it's easy, especially as long as Gaia is taking data.)
So there's a decent chance it could sneak up on us (modulo all the accretion stuff, of course!) for all the help neighboring stars give.
The next effect that comes into play would be on the planets of the solar system itself, and this will largely be due to tidal forces.
Bodies in free fall don't feel gravity, but do feel differences in gravity, but these difference -- the tidal forces -- scale as the inverse cube of the distance, so only get important as the SMBH gets close. How close? Look at the Hill Spheres of the Sun and a SMBH, though this only makes complete sense if there's a circular orbit involved.
A good-enough approximation can be made by looking at the distance where the change in the SMBH's gravitational attraction from one side of a planet's orbit to the other is the same as the Sun's attraction. (This results in complete disruption, of course.)
The formula (where mass in measure in solar masses and the distances are measured in planetary orbital radii) is R3/r3=2M, where R is the SMBH's distance, r is the planet's orbital radius, and M is the SMBH's mass measured in solar masses. Since, ex hypothesis, M is about 10,000,000, so R is about 300. So the SMBH's tides disrupts a planet's orbit at about 300 times the orbital radius.
So when the SMBH is 300 times further away than Pluto, it will have a major disruptive effect on Pluto's orbit, and likewise 300x for the other planets. Pluto is about 40 AU away, so at 12,000 AU distance the SMBH would completely disrupt Pluto's orbit -- that's about 0.2 light years.
Note that this is the distance for a complete disruption of the planet's orbit. There would be a significant effect at 10 times that distance or around two light years. (The less tightly bound outer planets are affected first. The Earth's orbit isn't disrupted until it gets within .005 ly and the effects are small at a tenth of a light year.
But we've been very good at measuring planetary orbits for centuries, so even a century ago, we'd have been able to detect the SMBH's tidal effects on planetary orbits while its still many light years away and hence many centuries away.
Maybe it won't be so good about sneaking up on us after all!
The fancy effects -- time dilation, frame dragging, etc., are only significant a few Schwarzschild radii away, which for a 10,000,000 solar mass SMBH, would be a few AU. These effects would definitely not come into play until long after the SMBH had forced itself on our attention by other means.
It depends how massive it is. Let's say it comes perpendicular to the solar system and there is something between in us that blocks our view, or we are not watching for whatever reason you choose.
For simplicity let's assign that gravitational pull from the Sun as 1 over distance of 1 astronomical unit. We would feel the same pull as the pull from the sun according to inverse square law. If your super massive black hole is as massive as black hole in the center of our galaxy my calculations goes like this:
Sagittarius A* 4.3×10^6 Solar masses
1 = 4.3×10^6 / x^2 x = (4.3×10^6) ^0.5 = 2073 AU ~ 0,033 Light years so coming with you at 1% speed of light it will swallow us in 3.3 years
If we take largest super massive black hole TON 618 6.6×10^10 solar masses the distance would be 812,403 AU ~ 13 light years and it will swallow us in 1300 years.
All this assumes that we are not ripped apart way before it, or thrown in intergalactic space.
I assume you mean the moment we could notice the black hole without being able to observe the black hole itself. Which means that in your scenario we can't use gravitational lensing or other optical or similar ray-kind effects. So if the hole itself would not be observable, but only the gravitational effects, the best indication would be that time slows down relatively in fields with high gravity. This would lead to the effect that the rest of the Universe would seem to speed up. This could be observed and this observation could lead to the realization of the existence of the said gravitational object.
If you wanted to know when we would feel the black hole, the answer would be: When it tears the world apart, which is, in my humbling opinion, not that great of a scenario.
EDIT: As noted in the comments, i need to add the answer to the question about how much time humanity would have after the discovery: And sadly, the answer can't be properly defined. Because of the strong distortion of time in such gravity, it's nearly impossible to define the exact time as relativly this timespan varies. Which has the advantage for you as an author that you can define the timespan. For every timespan the event would arguably valid.
