Tbh, this first time meeting and suddenly the ML being flirty and the FL start to have feeling for him is not really a great opening, I wish that this could open if the ML and the FL has a deeper connection in someway than just the "princes" tag that they have.
The FL feeling is reasonable since this is the first time she being interested in by another male, but the fact that the ML has to be so flirty is ick me so much. He just being too self-confidence. This whole thing could be better if the FL has a crush or just have a sight on him, when she first discover him doing something that hit her. Or the ML approach her but in more discrete way and at the end of the chapter, she find out that he likes her, this would be way cuter.
Or maybe this is just a shoujo and the demographic is girl so they make the ML someone extra flirty and handsome.