The spindle appears to be a 1/2'' solid rod with no external thread or internal taper. This is bad.
I had a Darra-James lathe of similar vintage that had the same deal. You will probably need to search Ebay for a drive center and face plate.
I don't mean to be negative, but it may be easier to just buy a newer lathe. Pretty much any lathe that is at least 10-15 years newer (i.e., ~1960+) would be much easier to work with. Alternatively, you might be able to find a replacement spindle that would fit and have more modern connections.
If you stick with this lathe, you will want to obtain a step pulley for the motor that matches the headstock pulley(s). This is actually a pretty standard part and you should be able to find one. You'll want to be sure the bore is the right size for your motor. This may be a match?