If Lynched, you return to your seat, and the game immediately goes to the night phase.
Player guide
As the Prince you main advantage is that you can not be lynched. This however does mean that you necessarily should let yourself be lynched, since that will mean the town will miss a day of lynchings that should ideally be used on a werewolf. As with every blue power role you can think of outing yourself either early to avoid having the town waste a round of lynching on you or mid to late game when the town wants to know who can be relied upon in the hunt for the last werewolves. As always outing yourself can attract the werewolves and make you a target but perhaps lesser so for the Prince than for the blue power roles with more active abilities.
Last and not least at all, but actually more first and foremost remember to be awesome and flamboyant. You are the Prince after all.
Long epic speeches are especially encouraged.
Backstory
These commoners and their petty problems, always whining to those in power to come and save them.Luckily for them, I am here. Born to inherit the kingdom and protect its people.
I am here to rescue the people from the terror in the night.
I can even manage this while enjoying a nice meal and a good glass of wine. And then get the hell out of this shithole village.
Innkeep bring me a bottle of your best red...