3D core-shell structures with excellent synergistic effects were synthesized by a combination of simple anodization and solvothermal methods and have great potential for application. In this work, we firstly generate copper hydroxide on the surface of copper foam (CF) by anodization, secondly, convert copper hydroxide into 2D Cu-BDC (copper (II)-1,4-benzenedicarboxylate) by solvothermal method, and thirdly, by solvothermal reaction, NiCo layered double hydroxide on the surface of Cu-BDC (CF@Cu-BDC@NiCo-LDH). The 3D core-shell structure of the CF@Cu-BDC@NiCo-LDH electrode provides excellent electrolyte transport and ultrafast electron transfer, providing an ultra-high specific capacity of 374.17 mAh·g−1 at 1A·g−1 and capacitance retention of 84.92% after 5000 charge/discharge cycles at 10 A·g−1. After 6000 charge-discharge cycles, the constructed asymmetric supercapacitor (ASC) has a maximum energy density of 59.56 Wh kg−1 and a maximum power density of 6397.97 W kg−1, with a capacitance retention rate of 89.4%. Furthermore, during the oxygen precipitation reaction (OER), CF@Cu-BDC@NiCo-LDH showed abundant active sites. It had a Tafel slope of 34.2 mV dec−1 and had an ultra-low overpotential of 306 mV at 100 mA cm−2. As a result, this research lays forth a realistic technique for designing and fabricating three-dimensional core-shell nanoarrays with different electrocatalytic activity and energy storage features.