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We study the minimum Monitoring Edge Geodetic Set (MEG-Set) problem introduced in [Foucaud et al., CALDAM'23]: given a graph G, we say that an edge is monitored by a pair u,v of vertices if all shortest paths between u and v traverse e; the goal is to find a subset M of vertices of G such that each edge of G is monitored by at least one pair of vertices in M, and |M| is minimized. In this paper, we prove that all polynomial-time approximation algorithms for the minimum MEG-Set problem must have an approximation ratio of Ω(log n), unless 𝖯 = NP. To the best of our knowledge, this is the first non-constant inapproximability result known for this problem. We also strengthen the known NP-hardness of the problem on 2-apex graphs by showing that the same result holds for 1-apex graphs. This leaves open the question of determining whether the problem remains NP-hard on planar (i.e., 0-apex) graphs. On the positive side, we design an algorithm that computes good approximate solutions for hereditary graph classes that admit efficiently computable balanced separators of truly sublinear size. This immediately yields polynomial-time approximation algorithms achieving an approximation ratio of O(n^{1/4} √{log n}) on planar graphs, graphs with bounded genus, and k-apex graphs with k = O(n^{1/4}). On graphs with bounded treewidth, we obtain an approximation ratio of O(log^{3/2} n). This compares favorably with the best-known approximation algorithm for general graphs, which achieves an approximation ratio of O(√{n log n}) via a simple reduction to the Set Cover problem.