Can we use reductions to design approximation algorithms for NP-hard problems? Let us say that I have a problem $P(n)$ that I need to solve (where $n$ is the size of the input of problem $P$). I used a polynomial-time reduction from a known NP-hard problem $Q(m)$ (where $m$ is the size of the input of problem $Q$) to $P(n)$ and showed that $P(n)$ is NP-hard. The time complexity of the reduction is $O(n^k)$ for some fixed $k$.

Can I use a known approximation algorithm for problem $Q(m)$ that is $O(f(m))$-approximation for problem $Q(m)$ to design an approximation algorithm for problem $P(n)$ that is $O(g(n))$-approximation for problem $P(n)$?
Yes, but you need an approximation-preserving reduction in the opposite direction.