Car hit by a lightning strike In Griffiths' Introduction to Electrodynamics, at the Electrostatics chapter, in particular, in the conductors section, he says this after the stating that within an empty cavity surrounded by a conductor material, the electric field is 0:

"(...) This is why you are relatively safe inside a metal car during a thunderstorm - you may get cooked, if lightning strikes, but you will not be electrocuted".

So, in a conductor, if I bring (for example) a positive charge q outside the conductor, induced negative charge will appear at its surface, negating the positive charge, killing off the field of q for points inside the conductor.
My question is: how fast does such process occurs in order to negate the charge of a lightning strike, "protecting" me inside the cavity (interior of the car)?
 A: If a container has a conducting surface it forms a Faraday cage.. There is a related answer here.
A car, if it is not plastic, is a metalic enclosure on rubber wheels. To be struck by lightning, ionizations paths between the charges on the ground and the opposite in the clouds form , and current flows discharging the energy. See this video of a car hit by lightning . Obviously the energy was not discharged inside the car, which as a Faraday cage did not offer inside  the least resistance path for the current to flow to the ground.

So, in a conductor, if I bring (for example) a positive charge q outside the conductor, induced negative charge will appear at its surface,

yes,

negating the positive charge, 

this is confusing.
What happens is that the surface of the conductor will become negative .The field of the positive charges stay within the conductor material ( or flow to the ground if grounded) and cannot penetrate inside a metallic enclosure, a Faraday cage, and thus no current can be induced inside the car for the lighting current to flow.
It is Gaus' law that does not allow field lines within a conducting enclosure. See "inside a sphere of charge"
A: The car dos not "repel" the lightning strike, as you seem to be picturing. Yes, it may actually separate the charges in the car's metal surface. But, say, the car has 5 positive particles and 5 negative. Even though they might be separated in different parts of the surface, the resulting charge in the car is still zero. Therefore, it cannot "repel" the lightning strike. 
What does happen is that the electrons in the lightning stick to the car's surface, because the metal in it is a good conductor. That's why you're safe in the interior of the car, because the electric charges of the lightning strike cannot reach you.
A: You are forgetting how lightning strikes are formed. When the electric difference potential, between a charged cloud and the ground, is greater than the dielectric constant of the air, a current goes down from the negative pole (Cloud) to the positive pole (Earth). So, what he is trying to say is that the current will travel along the conductor and because the electric potential is zero, there cannot be any charge on the internal surface of the conductor.
A: The reason a person in a car does not get hit by a lightning strike is because the roof and walls of the car present a path of less resistance than the interior of the car. 
For the same reason, a person could also be protected from a direct strike by a nearby tall conductive object - not a Faraday cage, but it would not be as reliable as a car and the person would have much greater exposure to light, heat and EM field produced by the current of the strike.
Your question is about the exposure of a person in a car to the EM field. It should be relatively low, but not zero, since a car has windows. The effect of the finite speed of the electrons cancelling the field should be negligible.   
