MV Sea Princess is one of my four favorite cruise ships (the other three are her identical sisters). You have chosen a real winner!
I don't know exactly what they have in the "Teen Zone" but I do know that most, if not all, of it is included in the fare. There might be a nominal charge for some activites such as video games (I seem to recall some discussion of people buying a "game card" that gave a certain number of plays on one of my past cruises, but I don't know the details). I do know that "PTC" ("Princess Teen Club") does have a lot of activities that teens seem to enjoy, though -- the only time when I have see teenagers just "hanging out" around the ship has been late at night when the activities were over.
It's rare to find a Princess cruise that does not have some young adults. If your 20-year-old joins in activities like the pool games & such that really do target young adults, he'll find plenty of others in his age bracket.
Have a gret cruise!