Since it's 120 volts, you won't need an external contactor for safety.
One of the contactor sets is make/break, with no crossing of the line(s). Use that for your 120 volt hot wire.
Bring the 5 & 8 out of the motor, to the drum switch, and have the drum switch swap them around, then send them back to the motor.
Yes, I am speaking theory, but that's because you must understand the theory, unless you like breathing ozone and replacing drum switches.
Until I copy your image, and number the contacts, this is the extent of my help
The 3,6,9 contacts are for your 120 volt hot line. Attach the incoming line to 6, and the motor to 6 or 9.
The others will swap your 5 & 8 wires. From the motor, attach 5 & 8 to 4 & 5, and attach the return wires to 7 & 8.
If there's a standard numbering scheme for drum switches, I didn't find it
You have adequate screws, it seems. But if you need more, most likely they are 32 tpi. I.E. 6-32 or 8-32 or 10-32 (3/16-32).
If you were to install a 240 volt motor, you would need to break both of the conductors. That would be accomplished with a "120 volt coil" 2 pole (or more) contactor that is adequate for the intended motor. Instead of using the 120 volts to power the motor, the 120 volts would be used to control the contactor.
Regarding "what sort of plug gives 120/240 volts?", there are 4 pole 120/240 volt "clothes dryer" plugs and receptacles, which are NEMA 14-30 series good for 30 amps.
Any less amps and you are into Twist Lock territory, such as NEMA L14-20 series.
As long as the cord supports the ampacity of the over current protection (circuit breaker?), the capacity of the plug/receptacle can be equal to, or heavier than the current draw.