Starting cap is bad. Cheap if you can find one. Do you have a parts list for the lathe? A capacitor is just a high capicity battery that can dump its charge really fast to kick start the motor. I have used a grinder like that for years. A little pull on the belt, then let go really fast and you are in business.
Yep, should be a little cover hanging off the side of the motor held by a couple screws. Under that cover ought to be a cylindrical shaped do-dad. You did unplug the thing, right? Take the cylinder thing out and find one just like it ('cept if it's cooked, you don't want a cooked one). Take it with you and a motor place ought to be able to find a match for it.
I got to go with the start cap. too. That puff of smoke you said was white in another post. That's the steam from the juice in the cap. We used to blow those up for fun. If you run too much juice into them, the fluid boils and it'll pop. If it has the two legs down, you can get the can to hit the ceiling. Lots of fun when you don't have to pay for them.
Anyhoo, a good hardware store may have them. I get mine right down the road. About $5 list time for the one on my BS.