If lower bound of a problem is exponential then is it NP?
No. For example, the halting problem has an $\Omega(2^n)$ lower bound, but it is not in NP (since it is not computable).

The nondeterministic time hierarchy theorem shows that any NEXP-complete problem is another example (with $2^n$ potentially replaced by a smaller exponential function $c^{n^\epsilon}$).

NP is an upper bound on the complexity of a problem.