The primal feasible point gives you a bound for the optimum value of the dual. Since you have a quasi-optimal primal point, and there is no duality gap, the primal function value at this point is also a good approximation for the optimum value of the dual problem.
But I can't see how this helps you to find a quasi-optimal solution to the dual. I guess it doesn't.
Edit: I just heard of a process calling "pricing out" the primal. Given a basic feasible solution to the primal, define z'=cb'B^(-1), where B is the basis associated to this solution and cb the corresponding basic primal cost. When the primal solution is optimal, the same happens with the dual. The thing is that when the primal solution is quasi-optimal, it turns out that the dual solution is quasi-feasible. I think this is the best notion to associate a dual solution to a primal one.