This is a very interesting question. However, in order to answer this question properly, one needs to take into account where this prince lives, as different cultures have different expectations as to how a prince is to behave. In order to answer your question, I am therefore assuming that the prince in question is living in the Western world, perhaps in Europe, where there is still a considerable number of active monarchies.
A key problem princes often face is the fact that the public feels that they are living a life of luxury, far removed from the common people, and that they are not doing any proper work. A prince is waiting to become king but isn’t a king yet. Neither does he have a proper career to bridge the time until he becomes king. Therefore, the general public could easily perceive a prince’s life as hedonistic, without any proper responsibilities or purpose.
In order to combat this perception, and in order to improve relations, it is important for a prince to lead a lifestyle that allows the general public to see how hard-working they actually are. This is why many princes have joined the armed forces, and in some cases even fought in war zones, such as Prince Harry, a British Prince, who went to war in Afghanistan.
Another way of keeping good relations is to show the general public that the prince cares about his people and seeks to improve life within the duchy or kingdom. This is the reason why many members of royal families engage in charitable work, as this allows them to raise attention to important causes and show their support for these. Prince William, second in line to the British throne, supports a great number of charities, such as homeless charities and charities dealing with mental health issues.
By making it clear that he understands the problems common people face in life, a prince will appear a lot more approachable and understanding. This will make him come across as humble and charitable, which in turn will be very helpful with regard to maintaining good relations with his people.