A sink-or-swim situation is one in which we must save ourselves by our own means or else fail. The image is that of a person thrown into the water without a life preserver; he or she must swim or drown.
Succumb or succeed, no matter what, as in Now that we've bought the farm, we'll have to make a go of it, sink or swim. This expression alludes to the former barbaric practice of throwing a suspected witch into deep water, often weighted down. In case of sinking, the victim died; in case of swimming, the victim was considered in league with the devil and therefore was executed. A related idiom,float or sink, was used by Chaucer in the late 1300s; Shakespeare had the current form in 1 Henry IV (1:3): “Or sink or swim.”