The remaining sophomore-level math courses would be multivariable calculus and differential equations. He would probably go into junior level courses in his second semester. These would be courses like proof-based linear algebra, abstract/modern algebra, and real analysis (not necessarily all at once).
Yes, for applied math, it depends on what area of application. For (theoretical) computer science, he would want to take some CS courses and then the algorithms/complexity and theory of computation courses (which may be in the CS department. For economics, he would take the math-heavy intermediate economics courses (with a calculus prerequisite; some department offer courses using multivariable calculus), econometrics, and mathematical economics courses as well as calculus-based probability theory.
For psychology, he may find good knowledge of statistics to be helpful, particularly at the graduate level.