You’ve got a 120VAC single phase AC induction motor here. Fractional HP motors are not typically “capacitor start” but I can’t tell that without a picture of the whole motor. A capacitor start motor will typically have a cylinder like metal bulge on it, under which the starting capacitor is located. Assuming no bulge, a Grainer type controller would probably work.
However the one you picked is not the best choice. The problem is that Grainer controller has a max current rating of 2.5amps, whereas the motor has a current rating of 4.6amps. This means starting the motor might burn up that Grainer controller.
If you wanted to experiment, a 600watt light dimmer switch from your local hardware store might work as some others have suggested in this post. A 600 watt dimmer is good to 5amps. The problem with dimmers is they’re usually designed for resistive loads like light bulbs. An AC motor is an inductive load that causes some lag in the current and this can mess up the Triac in the dimmer.
I like Hardwood’s treadmill motor idea myself. Treadmills use DC motors instead of AC motors. AC induction motors are designed to run best at a fixed constant RPM. That is because their speed is a really a function of the AC power line frequency (60Hz) and not the voltage. When you just cut voltage by using a dimmer, the motor still wants to run at 1725 rpm (in this case). However with less voltage, the motor makes less torque and it slows down due to the loss of torque. You get the illusion of speed control because you’re losing torque.
The DC motor in a treadmill would be a good match here since they have high starting torque and their speed can be controlled without a loss of torque. You can buy low cost DC motors from places like www.surpluscenter.com
for under $40, but they don’t come with a speed controller. That’s why I really like Hardwoods suggestion! You can get a free motor AND free speed controller from people adverse to exercise. As hardwood says, it won’t be long now before they start putting them along side the road!