A new copper-based metal-organic framework [Cu2L(H2O)2]·5DMF·2H2O (ZJNU-56) has been solvothermally synthesized using a custom-designed asymmetric rigid bent diisophthalate ligand, 5,5'-(1-amine-naphthyl-2,4-diyl) diisophthalic acid (H4L), and structurally determined by single-crystal X-ray diffraction. ZJNU-56 features a three-dimensional (3D) open framework incorporating three different types of metal-organic cages and two distinct types of one-dimensional channels. With a moderate BET surface area of 1655 m2 g-1, optimized pore structure, and functional sites (open copper sites and Lewis basic amine groups) on the cage surface, ZJNU-56 after desolvation exhibits highly selectively adsorptive separation of C2H2 and CO2 over CH4 under ambient conditions. At 298 K, the predicted IAST selectivities are 35.7-72.9 for an equimolar C2H2/CH4 gas mixture and 6.8-7.0 for an equimolar CO2/CH4 gas mixture at pressures varying from 1 to 109 kPa, respectively, which are among the highest reported to date for copper-based MOFs.