Commercial metathesis polynorbornene is used for the fabrication of high-damping coatings and bulk materials that dissipate vibration and impact energies. Functionalization of this non-polar polymer can improve its adhesive, gas barrier, and other properties, thereby potentially expanding its application area. With this aim, the post-modification of polynorbornene was carried out by inserting ethylene–vinyl acetate–vinyl alcohol blocks into its backbone via the cross-metathesis of polynorbornene with poly(5-acetoxy-1-octenylene) and subsequent deacetylation and hydrogenation of the obtained multiblock copolymers. For the first time, epoxy groups were introduced into the main chains of these copolymers, followed by the oxirane ring opening reaction. The influence of post-modification on the thermal, gas separation, and mechanical properties of the new copolymers was studied. It was shown that the gas permeability of the copolymer significantly depends on its composition, as well as on the amounts of hydroxyl and epoxy groups. The developed methods efficiently improve the barrier properties, reducing the oxygen permeability by 15–33 times in comparison with polynorbornene. The obtained results are promising for various applications and can be extended to a broader family of polydienes and other polymers containing backbone double bonds.