Is minimax problem NP-Hard when the inner problem is NP-Hard ?
The answer to your question is "no", although for natural problems that you might be thinking about, it may be a good heuristic that if the inner problem is NP hard, the whole problem is probably hard as well. But you can come up with the following contrived counterexample.

Let $X = \{0,1\}$ and $U = \{0,1\}^n$, and let $\phi$ and $\theta$ be two 3SAT instances with the promise that exactly one of them is satisfiable. Let $f(0,u) = \phi(u)$ and let $f(1,u) = \theta(u)$. Then the minmax value is always 0, so computing it is in P, but for any fixed $x$, the problem is 3SAT and is NP hard.