cast = to throw
to cast one's lot with = to associate oneself with and share the good or bad fate of
whatever adj. = any......that
-any woman that may cast her lot with mine
subject = woman
verb = may cast
object = lot*
should one ever do so = if one should ever do so.
lot - noun
An object (usually a piece of wood) used in a widely diffused ancient method of deciding disputes, dividing plunder or property, selecting persons for an office or duty, etc., by an appeal to chance or the divine agency supposed to be concerned in the results of chance. The ‘lots’, each bearing the special mark of one of the competitors, were placed in a receptacle (in Homeric Greece a helmet); according to Greek procedure the vessel was shaken, the winning lot being that which was cast out (thrown out) first; in Scandinavia the winning lot was drawn out by an uninterested person.
from the Oxford English Dictionary