If Rubinia Soulsinger stops being tapped before its ability resolves — even if it becomes tapped again right away — you won't gain control of the targeted creature at all. The same is true if you lose control of Rubinia Soulsinger before the ability resolves.

Once the ability resolves, it doesn't care whether the targeted permanent remains a legal target for the ability, only that it's on the battlefield. The effect will continue to apply to it even if it gains shroud, stops being a creature, or would no longer be a legal target for any other reason.

If Rubinia Soulsinger leaves the battlefield, you no longer control it, so the duration of its control-change effect ends.