The ring-opening copolymerization (ROCOP) of epoxides and cyclic anhydrides is a promising method for the synthesis of new polyesters with various polymer properties. Among previously reported metal catalysts for ROCOP, the Schiff-base complexes have gained significant attention because of their ease of synthesis and modification. In this work, zinc and magnesium complexes containing Schiff-base ligands with different alkoxy side arms [-(CH2)2O- and -(CH2)3O-] were synthesized and shown to have a cubane metal core by X-ray crystal structures. All complexes were studied in the ROCOP of cyclohexene oxide (CHO) and succinic anhydride (SA) in toluene at 110 °C. The zinc complex having a shorter side arm is the most active catalyst for copolymerization, giving poly(CHO-alt-SA) with narrow dispersity and negligible ether linkage. On the other hand, magnesium complexes were not active because of the formation of stable carboxylate species. The detailed analysis of polyester obtained from zinc complexes unexpectedly revealed three different types of polymer structures occurring at different polymerization times. Cyclic polymer was generated at the beginning by intramolecular transesterification of the alkoxy side arm, giving a low-molecular-weight polyester. At higher conversion, cyclization diminished, giving just a linear polyester but with minor competitive formation of higher-molecular-weight polyester having cyclohexanediol as an end group. On the basis of a thorough understanding of the polymerization mechanism, the desired cyclic poly(CHO-alt-SA) was successfully synthesized using a low monomer/catalyst ratio.