Most asteroids large enough to cause an "extinction event" have been found and future orbits calculated for hundreds of years.
What might hit are smaller asteroids and long period comets. There is almost nothing we could do with a large long period comet. While we might get several years of warning, there is almost nothing that could be done.
Smaller asteroids we would get no warning on most of the time. There is a lot of sky, and only a tiny fraction of it is searched by something big enough to see a "city killer" a month away. A week's warning is more likely, even a day's warning is less than 50%. The most likely first warning would be a bright spot in the sky as the asteroid starts to hit the top of the atmosphere... Seconds before impact.
To get warning of city killers would require putting up some specialized sky survey satellites. Unlikely, in today's political environment.