Exponential reduction vs Polynomial Reduction
In fact, depending on the class you want to reduce to, you need to be careful about the reduction you are doing.

For instance to preserve the class $P$, you need to do $LOGSPACE$-reductions.
For the class $NP$, you need $P$-reductions.

I'm also afraid that you reverse the reductions: to show that a problem $B$ is in $NP$, you need to reduce it (polynomially) to a problem $A$ in $NP$. Meaning that if you to how to solve $A$ in $NP$, then you can solve $B$ by reducing it to $A$.

So to sum up, if $B\leq_P A$ and $A\in NP$, then $B\in NP$. Indeed the NP algorithm is clear: perform your reduction (polynomial time) and then solve the instance of $A$ you reduced to (in $NP$).

However, if the reduction is exponential, then we cannot say anything about $B$, except that it is in $EXPTIME^{NP}$, i.e. you can solve it in exponential time with an $NP$ oracle. It is in fact the same that just saying $B\in EXPTIME$, since the NP oracle does not add power.