This is an answer to the two more questions asked in the comments. I started it out as a comment, but got tired of the space restriction....
- I am not certain, but you are right, $NE(X)$ for this $X$ has rank at least $2$. I think that depending on the actual cover you choose the Picard number of $X$ may or may not be larger than $2$. Here is how you can study this: I think the key is to understand the Picard group of the fibers. It seems that in the general case all of these Picard groups should have rank $1$. If that's so, then using cohomology and base change (Hartshorne, Chapter III, Section 12) you can prove that then the Picard number of $X$ is $2$. In that case $NE(X)$ can be of only one type.
If the Picard number of $X$ is larger than $2$, then it gets more complicated. For a description of what can happen for a K3 you could look at The cone of curves of a K3 surface and The Cone of Curves of K3 Surfaces Revisited. I think that if you identify the relative cone of the fibration, you have a good chance at figuring out $NE(X)$ or at least enough about it so you can do whatever you need this for. On the cone of CY manifolds there are some results by Wilson, Morrison, Kawamata, Totaro.
- Yes, the Kähler cone is the dual of $NE(X)$ in the sense you wrote it. This follows from Kleiman's criterion for ampleness: If $D\subset X$ is a divisor on a smooth projective variety, then it is ample if and only if $D\cdot\sigma>0$ for any $\sigma\in\overline{NE}(X)\setminus\{0\}$. This can be found in pretty much any book dealing with higher dimensional (birational) geometry, for example Lazarsfeld's Positivity in Algebraic Geometry I.
