I think there is no ramification in the case you are interested in. 

The surface $\tilde X$ is the minimal ruled surface $\mathbb F_2$. The map $\tilde X \to X$ contracts the $-2$-curve $E$ to a point and maps the fibers of the ruling of $\mathbb F_2$ to lines in $\mathbb P^3$. So at every point of $E$ the map has differential of rank 1 and the kernel of the differential is the tangent space to $E$, because $E$ is contracted.

If $E$ intersects $\tilde C$ transversally, 
by the previous remarks, $\tilde C \to C$ has non zero differential at $x$.