Given that A reduces to B in $O(n^2)$ and B is solvable in $O(n^3)$, solve A
You have a reduction, but you don't know what size the new problem will be. 

Say you have an instance of problem A of size n. And you can reduce it to an instance of problem B in time O (n^2). If you can reduce it to an instance of size $O(N^{1/2})$ then you are fine: You took O (N^2) for the reduction, and $O(N^{3/2})$ to solve the instance of B. 

If you reduce the instance of problem A to an instance of size O (n log n) of problem B, then you will take O (n^3 log^3 n) to solve the instance of B. 

So the time for the reduction and the time for solving B are not enough. You need to know the size of the new problem as well. (Of course you can't create an instance greater than O (n^2) in O (n^2) time, so the worst case is O (n^6), but it could be much better).

The answer that was given (O (n^3)) is definitely wrong. It doesn't follow from the information you were given.