The crime was captured on grainy video footage which could leave no one in any doubt as to who had pulled the trigger. The prince certainly did commit murder and it is highly unlikely that the Saudi state would have executed a prince if there was any potential doubt lingering over the details of the crime.
Prince Turki bin Saud al-Kabir pleaded guilty to the crime. Maybe he did so assuming that he would receive leniency because he was a member of the royal family. However, he must have known that death was one possible sentence if the victim's family refused the blood money. We should accept that his confession was not coerced.
The Saudi Royal family is not known for punishing it's own members, and they made their decision because of the overwhelming popular decision. The Saudi Royal prince did commit murder or he would not have angered quite so many people. There was enough evidence for them to make their decision.
No, there was not enough proof because in countries like Saudi Arabia proof does not matter as much. In the US, everyone is innocent until proven guilty, but that is not the case in every country. In fact, many countries believe you are guilty until proven innocent. It is just a different way of judging someone.