Does this problem map to the Set Packing problem? Let $G(m,n)$ be A bipartite graph $G$ with paritions $m$ and $n$ with the property that partition $\mathit n$ has two types of nodes (type1 or type2). 

Given $G(m,n)$ and $k \in \mathbb Z+$:

Does $\mathit S\subset \mathit m$ where |$\mathit S$| $=$ $\mathit k$ exist so that none of $\mathit S$'s nodes are adjacent to nodes of type1, but all type2 nodes in $n$ are adjacent to nodes in $S$? ($S$ is a subset of the $m$ partition).

I can see a solution is easily verified by checking that each type2 in $n$ has a neighbor in $S$ and that |$S$| $=$ $k$. This means the problem is in NP.

To reduce the problem of Set Packing to the above, I am thinking to consider the of all of $m$ nodes's neighborhoods as subsets. But from here I am not sure how to proceed with the problem mapping.
You cannot reduce set-packing to your problem, unless P=NP.

There is a linear-time algorithm for your problem.

Remove all vertices of type 1 and all their neighbors. Let $m'$  (resp. $n'$) be the set of remaining vertices in $m$ (resp. $n$). 
If $|m'| \ge k$ and the neighborhood of $m$ contains all the vertices in $n'$, the answer is yes.

If any of the above two conditions is not satisfied then the answer is no:


If $|m'| &lt; k$ then any set $S \subseteq m$ with $|S| \ge k$ contains a vertex that is adjacent to some vertex of type 1 in $n$.
If there is a vertex $v \in n$ of type 2 that is not in the neighborhood of $m'$, then every vertex $u \in m$ that is adjacent to $v$ is also adjacent to a vertex of type 1 in $n$.