Determine shortest path in a 4x5 grid graph
Since your graph is a DAG you can topologically sort $V$ and solve your problem via dynamic programming by considering the vertices in reverse topological order.

For each vertex $v$, let $d(v)$ be the distance from $v$ to $v_{n,m}$.
Clearly, if $v = v_{n,m}$, then $d(v)=0$.

For $v \neq  v_{n,m}$, you have:
$$
d(v) = \min_{(v,u) \in E} \{ w(v,u) + d(u) \},
$$

where $w(v,u) \in \mathbb{R}$ is the weight of edge $(v,u)$ in $G$.

Both the topological order of $V$ and all the quantities $d(v)$ can be found in $O(|E|)$ time. Therefore the overall time required is $O(|E|) = O(|V|) = O(nm)$.