IIRC, starter rope is a tight braided dacron cord. Has good abrasion resistance and doesn't stretch. When needed any rope of a similar OD will work. If you want a spare rope any hardware or lawn equipment store should have the right stuff. It's not expensive so buy 10 feet and have enough for next time.

As far as pulling apart your recoil starter, as OBG said it is "intersting" the first time. Different manufacturers make different mechanicals, some are easy. Can do one of those in 5 minutes, others make you swear that you are not going to touch another one of these things ever again.