What does face the final curtain mean?

It means to look at the end of one's life; to face death. When the play ends, there may be 'curtain calls'; actors come out for a bow, and most of the time the whole cast comes out at one point for a bow. In theaters with full curtains, these curtain calls may be separated by the closing of the curtain. The conceit is that "we're all done, but, OK, you want to see us again because the performance was so brilliant, so we'll go ahead and open the curtain again for a bow." Then at some point the curtain comes down and there is no more applause; it's the end, the 'final curtain'.