Do you assume the ideal is homogeneous? No power of $S$ belong to $(X,S-1)$. 

If you assume that the ideal is homogeneous, then $S^m$ is in the ideal $(f,g)$ for $m=\deg f+\deg g-1$. Indeed all homogeneous polynomials of that degree are in the ideal $(f,g)$. This is a simple Hilbert series computation, just use the fact that the Koszul complex resolves $K[X,S]/(f,g)$ and use it to compute the Hilbert series of $K[X,S]/(f,g)$.