Proceed with fingers crossed! There are at least 3 different types of single-phase motor. Some motors have one capacitor, others two, and there is often a centrifugal switch as well.
Judging by the photo, this example is simple: a single START capacitor (only two wires leading to it), without a RUN capacitor, and probably without a centrifugal switch. If there is one, and perhaps it is faulty, should be found at the pulley end of the motor. It connects the RUN capacitor to the windings when the motor is stopped, and disconnects it once the motor is running. If the switch fails open, the motor won’t start: if it fails closed, the motor will overheat. (But I guess this motor is designed with a hefty run winding that’s left in circuit permanently, and there is no switch.)
The capacitor value(s) depend on the windings and other technical characteristics of the motor, known to the designer, but rarely shared with the public. One’s best hope is that the old capacitor is marked, but they aren’t always, and of course ancient markings tend to become unreadable. Have a close look, perhaps breathing on the can to see if a little condensation brings any markings out.
Otherwise, Chuck’s answer is a good place to start, even though it requires us to guess the power-factor at start, which isn’t the same as the motor’s power factor when running, and isn’t easy to measure. 0.8 is reasonable. So applying Chuck’s formula for this 1/6HP HP motor suggests a 5 microfarad capacitor.
The good news is that the value of the capacitor in this application isn’t too critical – only has to be near enough to start the motor. Too small is worse than too big, so I’d try anything between 5 and 15uF. If the motor won’t run (and the windings/switch are OK) try bigger.
It’s important to use a Motor Start Capacitor rather than any other type. They’re common as muck: this page is Mouser, plenty of other suppliers like Farnell, RS Components, etc etc. The capacitor must be rated for more than mains supply voltage, so 280Vac in the UK, which is usually well above 220V.
Is fixing an ancient motor is worth the cost and risk? By modern standards, old insulation has poor resistance to age, heat and wet. The photo shows the original wires leading to the capacitor are unacceptably dangerous, and they at least have to be replaced, and a grommet fitted. The interior wiring may not be much better, especially if the motor has ever got damp.
The risk of using a potentially dodgy motor depends on the circumstances: extremely unwise to leave one running unattended because it might catch fire and burn the building down. Assuming the owner knows what to look for and the motor doesn’t get hot, smell, or tingle, intermittent use on a small lathe in a dry workshop is probably OK provided the whole is competently earthed. A professional would insist on a Megger test.
Dave