Research into novel materials with tailored properties is essential to improve CO2 adsorption and activation, a fundamental preliminary phase in catalytic CO2 conversion processes. Here, we have employed density functional theory-based calculations to investigate CO2 activation over the pristine and Cu-decorated carbon-based two-dimensional material ψ-graphene and its hydrogenated forms, i.e. ψ-graphone (half hydrogenated) and ψ-graphane (fully hydrogenated). ψ-graphene is a metallic allotrope of graphene containing 5-6-7 membered carbon rings. Our study found exothermic binding of CO2 at all three materials (for both pristine and Cu-decorated materials), indicating spontaneous physisorption. The presence of a single Cu atom plays a significant role in increasing the activity of ψ-graphene towards CO2 activation. By decorating ψ-graphene with Cu atoms, the adsorption energy at ψ-graphene increases about three times, whereas no significant variation is observed on ψ-graphone and ψ-graphane sheets. We used Bader charge analysis and electronic density of states plots to further quantify the nature of the interaction between CO2 and these materials.