Correct. The word brim comes from men's hats, which have a brim to keep the rain from falling in your face. The brim is essentially the edge of the hat, and it is common in English to call the top edge of a cup the "brim." A famous line from Shakespeare says, "My cup runeth over" meaning that I am over-abundant in some particular way for the current moment. To be full to the brim with enthusiasm means that the enthusiasm has overcome you and is at it's limit; there cannot be more because it is full, so we say full to the brim with the emotion we are feeling at the time when it feels to be almost too much for us to handle.

Yes, the person is full of enthusiasm. If a cup is full to its brim (the edge) it cannot hold any more - it is as full as it can possibly be.