Can I buy a black used bumper and paint it in white by a dealer for my white car? Or do I only have to buy a white bumper or plain (without paint) original for further painting?
Yes, as long as there isn't any damage to it, you can do this. You don't even need to get it painted by the dealer. Most body shops (if their worth their mettle) can color match to your car. You'll need to allow them to pull the color code off of the car, then you're golden.
It is easier to get a clean, unpainted one for the body shop to paint, but they can do it. They'll have to scuff the paint which is on there, fix any blemishes, then paint it. If the paint which is on it is not in good shape (cracked or pealing), they'll have to strip it for you, then primer, then paint it. This would cost a little more in the preparation department, but it can still be done.