will it be very crowded?Yes. New Year is one of the main travel seasons of the year.
will most of the shop/restaurant open? especially the hot spring egg?
Good question. I think that quite a few shops and restaurants might be closed on January 1, but should re-open on January 2, but it is possible that most is open on January 1. Frankly, I am not sure.
all the hakone round course transport will still be first come first serve?
Yes.