Great question! I think one approach you could take is to think of the internal conflict that is going on within the "semibarbaric" king that we are introduced to in the first paragraph:
In the very olden time, there lived a semibarbaric king, whose ideas, though somewhat polished and sharpened by the progressiveness of distant Latin neighbours, were still large, florid, and untrammelled, as became the half of him which was barbaric.
This indicates a man with two halves, each in conflict with the other. The barbaric side, of course, represents man, as he wants to follow his natural inclinations, however, he has to balance this with the "civilised" side that cannot be as barbaric as he would like.
Of course, another external man vs society conflict is evident in the princess and her desire to do what she wants and have a lover rather than play the role that society has ordained for her. This is what results in the terrible decision that she must make at the end of the story. Because she has followed her own inclinations and fallen in love with someone privately, she has gone against society and her job as princess, for it is not for her to choose with whom she will marry and with whom she will fall in love.
Therefore, analysing the character of the King and the Princess yields two interesting conflicts between man and society that you can hopefully write about and analyse a bit further. Good luck!
To add to the excellent answer provided by accessteacher, I think another valid man vs society conflict would be the victim of the arena versus the arena itself. The arena could be connected to a value of their society, as it is the means of which they execute justice. The person in question, who is the accused is confronted by this societal value and the resolution of this conflict would be which door the person would choose. Also, another one, but one that is perhaps sketchy, would be the princess vs society. She, by finding out what is behind which door, is directly combating the supposed purpose of the arena, that is to give out divine judgement. By interfering with this process, she could be said to be conflicting against it.