As Ryan commented, there are maps of every degree from $S^2$ to itself, and so we cannot work with arbitrary maps. However, the Riemann-Hurwitz formula allows you to generalize to ramified covering maps (i.e. maps which are coverings after removing a finite number of points) of surfaces. Where the cover is ramified, you need correction terms, and these terms can be given explicitly in terms of how the sheets of the covering come together.
These two examples show that degree is only one aspect of how maps affect Euler characteristic. The Riemann-Hurwitz theorem is closely related to the Riemann-Roch theorem, and so the Grothendieck-Riemann-Roch theorem might be in some sense the best "generalization" of the Riemann-Hurtiwz formula, although I wouldn't not consider it a direct generalization in the usual sense.