Is DAG subset sum approximable?
It seems to me the pseudo-polynomial time dynamic programming algorithm for Subset Sum problem also works for this problem. For each vertex $v_i$, we compute the set $L_i$ consisting of all possible values of paths ended at $v_i$. Then, we have the recurrence relation: 
$L_i=\{g(v_i)\}\cup\{x+g(v_i)\mid x\in \bigcup_{j\in  prec(i)} L_j\}$. Following a topological order, all the $L_i$ can be computed in $O(Km)$ time, where $K$ is the total weight and $m$ is the number of edges.

I think the standard scaling-and-rounding can also be applied to derive a FPTAS.