The moon has been slowly moving away from the Earth for the past 4 billion or so years. It's current rate is 3.78cm/year and it is currently 384400 km from Earth.
So even if it reversed direction and started falling in at 1km a year, it would still take hundreds of thousands of years to smash into the Earth.
But what would cause this action? While the moon is significantly smaller than the Earth, it is still a very large mass to disrupt to such an extent. Most things that would be able to do so are going to be disrupting things through out the entire solar system. Maybe even changing the orbit of the Earth. Say a large rogue planet passing through, or a small star or black hole.
So there will be a lot more to worry about than the crashing of the moon into the Earth for quite a while. Ecosystems and the entire planet will be in upheaval. Quite possibly that the crust will become more active as well, increasing plate tectonic activity.
If we survived the climate change and the tectonic activity, then we might start thinking about moon and trying to decide if we can change it's trajectory or if we should plan on vacating our home, since we've been given an eviction notice...
So even changing the path to close in on the Earth at 1000km a year, we give us 300-400 years to prepare and try and get as many people off the planet as possible. The closer the moon got the more tectonic activity would increase as well as tidal actions, if we continued global warming and more ice melted, Oceans would rise and with higher tides, much of our coastal areas would become inhospitable to living, unless we created island cities on the oceans.
Our biggest problem would be fighting the panic since there is no way we will get even a billion people off this rock much less the billions. So social unrest will be rampant with all that entails. Our biggest problem for survival of such an event is really ourselves.