Well, thanks for noticing that for me. I'm new to big electric motors, didn't realize some required capacitors. Now the question is, which capacitors?.. I'm trying to find a real data sheet on that motor but I can't find one... I'll investigate some other motors that match my specs.
We're stuck at 110 volts residentially over here, so I'd need both a transformer and a phase converter. Speed control isn't that critical, but it would have been nice.
You don't need to add capacitors, "Capacitor start" means the motor is a single-phase induction motor that has a built-in capacitor to start it. Induction motors run at speeds defined by the supply frequency and load, so you can't speed-control them unless you use an inverter to vary the supply frequency.
To switch the motor on and off from an Arduino, use either a mechanical relay driven by a transistor, or a random-fire SSR. You can make an SSR from a triac, opto triac and a few resistors, but it's better to buy a ready-made one.