Not an expert,but I would think 1/3rd HP is a bit small for anything,never mind a lathe. Might be OK for a sanding disc or something like that. Might be better off looking for something in the 3/4 HP range. Most of the old Cromtom Parkinson [have I got the spelling right? ] motors could be reversed very simply and they went on for ever and a day.
I have some 50 years old and still going strong !
No sense in fitting a motor that,s too small,you end up with twice the work,having to replace it.
Think about it.