I couldn't book tickets in advance through the website. As I checked now the date are already booked.
So I want to know whether I can but tickets on premise to the cruise and to visit the crown of statue of liberty on premise.
As the official National Park Service website states clearly,
Due to its extreme popularity, access to the crown is by advanced reservation only. Statue Cruises is the official source for crown reservations as well as the company that provides ferry transportation to the island. All reservations must be made online or by telephone.
Crown reservations are not available at the ticket offices.
Crown reservations are not available on Liberty Island.