The correct answer is $\boxed{\text{A}}$.
The angular speed of an object is the rate at which it rotates around a particular point. It is measured in radians per second. To calculate the angular speed, we use the following formula:
$$\omega = \frac{v}{r}$$
where $\omega$ is the angular speed, $v$ is the linear speed, and $r$ is the radius of the circle.
In this case, we are given that the linear speed is $36 \text{ km/hour}$ and the radius is $100 \text{ m}$. We can convert the linear speed to meters per second by multiplying it by $\frac{1 \text{ hour}}{3600 \text{ seconds}}$. This gives us a linear speed of $10 \text{ m/s}$.
Now we can plug in the values for $v$ and $r$ into the formula and solve for $\omega$. This gives us:
$$\omega = \frac{10 \text{ m/s}}{100 \text{ m}} = 0.1 \text{ rad/s}$$
Therefore, the angular speed of the car is $\boxed{0.1 \text{ rad/s}}$.
Option B is incorrect because it is the linear speed of the car, not the angular speed. Option C is incorrect because it is 100 times the angular speed of the car. Option D is incorrect because it is 1000 times the angular speed of the car.