Is it possible to design an efficient approximation algorithm for one NP-complete problem based on Shor's algorithm? Is it possible to design an efficient approximation algorithm for an $\sf{NP\text{-}complete}$ problem based on reductions from Shor's algorithm? 

Are known any (classical) approximation algorithms for an $\sf{NP\text{-}complete}$ problem that uses Factoring as oracle?
There are ways to see that either the answer is probably no, or that the question means more than one thing and has a negotiable answer.   On the one hand, the PCP theorem says that many, but not all, NP-hard problems are still NP-hard to approximate.   The standard belief is that Grover's search algorithm, which gives you a quadratic speedup but no more than that, is the best quantum algorithm for the hardest NP-hard problems.  This leaves fairly little wiggle room to expect any quantum algorithm to have any special relation to approximation to NP-hard problems in general.

Some NP-hard problems are easier to approximate than the ones amenable to the PCP theorem.   However, the difficulty of approximation is then highly variable.

Meanwhile Shor's algorithm does something very specific:   It finds the period of a periodic function on the integers or on $\mathbb{Z}^n$.   This problem is also in the complexity class SZK, for example.   Maybe you could cook up an approximation problem to an NP-hard problem that lands you in SZK or period-finding, but I suspect that there aren't any known, natural examples of that.