As an undergraduate, you must take ALL of the available physics courses and their pre-requisite math courses. You must take as many advanced-level courses in classical and quantum mechanics, calculus, differential equations, mathematical physics, and you must develop a fluency in at least one computer language, usually FORTRAN.
There is no difference between you and a physics student, except that you will also take several advanced undergraduate courses in introductory astrophysics including stellar evolution, cosmology, planetary physics and galactic structure. You will take all of the lower-level calculus courses, and later take advanced courses in solving differential equations, fourier transforms, and statistics. In physics, you will study classical mechanics, relativity, nuclear physics, statistical mechanics and electromagnetism.
It is hard work, but astronomy is not just about counting stars through an eyepiece! In the last 30 years, the requirements for entry into the profession have increased to the point that we are now all just physicists applying physics to just another area of the world with different terminology.