We demonstrate high-power laser delivery exceeding 1 kilowatt through a 5-meter homemade anti-resonant hollow-core fiber (AR-HCF) at 1-µm wavelength. Laser-induced damage to the fiber coating and jacket glass is experimentally observed respectively for different incident laser powers from a few hundred watts up to nearly 1.5 kilowatts. The cladding microstructure of the AR-HCF is free of damage at the incident end when 80% of the 1.5-kW incident power is coupled in. The deviation of an incident laser beam from the core to the cladding causes no damage but only deterioration of the coupling efficiency. The potential of the AR-HCF for higher-power laser delivery is discussed.