There are two ways to look at this: (1) Mark distance measurements from 0 to 1 on the unstretched elastic, and then measure the fly's progress with respect to those marks, or (2) measure the fly's absolute speed to see if it ever exceeds the speed of the car by enough to catch up to the car.

Of these methods, offhand it seems that (1) is easier, so we mark the distance on the elastic, and then let's say the fly walks at "x" meters per second.

So the fly's instantaneous speed measured against the marks on the elastic is

x/(1+t)

where t is the time, in seconds, that have elapsed since the fly started walking.

The distance that will be covered by the fly after a seconds is the integral of x/(1+t)dt from 0 to a.

This integral is x ln(a+1)

For the fly to reach the car, we have

x ln(a+1) = 1

Solving for a, we see that the fly will reach the car after

a = -1 + e^1/x seconds

This gives a value of a for any positive value of x. So if the fly walks at the rate of .01 meters per second, for example, he will reach the car after -1 + e¹⁰⁰ seconds, which is about 2.68812E+43 seconds, which is about 10²⁶ times the age of the universe.

Never say never!