I have a deep cut above my upper lip on the outside and inside. No stitches were required. How long will my lip be swollen? Can I clean it with water

Deep lip cut. Whether stitches are 'required' is a matter of surgical opinion. Any deep cut on both the inside and outside of a lip which are in the same area must be assumed to be 'through and through' and therefore must be carefully evaluated--and likely closed with stitches. There is a misconception that 'stitches' are somehow bad and that 'avoidance' of them is best. This is bad reasoning.
Injury. The swelling will resolve in 2-3 days or even earlier. The wound will heal in 5-7 days. Lips, face is richly supplied with blood, for this healing is faster- unless there is infection. You said no stitches required, means you visited er? Keep your mouth clean, you can clean the wound with water or hydrogen peroxide.