The oxygen reduction (ORR) and oxygen evolution reactions (OER) in Zn-air batteries (ZABs) require highly efficient, cost-effective and stable electrocatalysts as replacements to traditionally high cost, inconsistently stable and low poison resistant Platinum group metals (PGM) catalysts. Although, nitrogen-doped carbon nanotube (NCNT) arrays have been developed over recent decades through various advanced technologies are now capable of catalyzing ORR efficiently, their underdeveloped bifunctional property, hydrophobic surface, and detrimental preparation strategy are found to limit practical large-scale commercialization for effective rechargeable ZABs. Here, we have demonstrated fabrication of a three-dimensional (3D) nickel foam supported NCNT arrays with CoNi nanoparticles (NPs) encapsulated within the apical domain (denoted as CoNi@NCNT/NF) that exhibits excellent bifunctional catalytic performance toward both ORR (onset potential of 0.97 V vs. RHE) and OER (overpotential of 1.54 V vs. RHE at 10 mA cm$^{-2}$). We further examined the practicability of this CoNi@NCNT/NF material being used as an air electrode for rechargeable ZAB coin cell and pouch cell systems. The ZAB coin cell showed a peak power density of 108 mW cm$^{-2}$ with an energy density of 845 Wh kg$_{Zn}^{-1}$ and robust rechargeability over 28h under ambient conditions, which exceeds the performance of PGM catalysts and leading non-PGM electrocatalysts. In addition, density functional theory (DFT) calculations revealed that the ORR and OER catalytic performance of the CoNi@NCNT/NF electrode are mainly derived from the d-orbitals from the CoNi NPs encapsulated within the apical dominant end of the NCNTs.