The pilot reported that on a night landing the airplane touched down smoothly on the main landing gear. During the landing roll, when the nose gear touched down, the airplane’s roll out became rough and bumpy. When the airplane came to a stop, the pilot and passenger discussed that they probably had a flat nose wheel tire. Subsequently, the passenger exited the airplane to examine the nose wheel tire and observed it to be flat. The pilot told the passenger to keep clear and then attempted to taxi clear of the runway. However, at a high-power setting, the airplane only moved a few feet, and the pilot elected to discontinue the taxi. The pilot was deciding what his next move would be when the passenger approached the airplane from the front. The pilot tried to warn the passenger but subsequently, the propeller struck the passenger and resulted in a serious injury.