Compute sum of edges in paths from source to target node?
A simple approach that should work w/out need for a hash table is as follows:
Let two boolean arrays $a,b$ such that

$a[u] =1 $ iff a path $1 \ldots u$ exists

$b[u] =1$ if a path $u  \ldots i$ exists


Both can be calculated in $O(V+E)$ time : $a$ with a simple BFS from $1$, and $b$ by a bottom-up approach of examining the vertices in the reverse topological order.
Now, you consider each edge of the graph $ u \rightarrow v$ and add $w(u,v)$  to the answer iff $a[u]=1$ and $b[v]=1$.
Because the graph is a DAG the two paths above must be disjoint thus their concatenation is a valid path from $1$ to $i$ and since we consider each edge once, there are no duplicates.