Yes- it is correct. In fact, plug some random numbers in one of those calculators you have laying around and see the effect of changing activation energy and temperature.
You are correct that an increase in temperature will increase the rate. That makes sense, right? More kinetic energy, more chances of collision with enough energy in the proper orientation?
Activation energy is just like temperature. Doubling the temperature doesn't double the rate. The activation is in the exponent with temperature, right? So no, doubling the activation energy doesn't cut the rate in half.
Be cautious of the inverse natural log in the Arrhenius equation.