Yes a woman can have her saggy breasts lifted without having breast implants done.
When you go to the surgery to get your breasts lifted the surgeon will perform a breast lift which removes excess skin and raise and reposition the nipple-areolar complex, and also reshape your native breast tissues which is in a way similar to a small breast implants in those that don't want the breast implants.
So when you go in for breast surgery ask for the breast lift to be done instead of the breast implants.