The classic Fenton reaction, which is driven by iron species, has been widely explored for pollutant degradation, but is strictly limited to acidic conditions. In this work, a copper-based Fenton-like catalyst Cu/Al2O3/g-C3N4 was proposed that achieves high degradation efficiencies for Rhodamine B (Rh B) in a wide range of pH 4.9–11.0. The Cu/Al2O3 composite was first prepared via a hydrothermal method followed by a calcination process. The obtained Cu/Al2O3 composite was subsequently stabilized on graphitic carbon nitride (g-C3N4) by the formation of C−O−Cu bonds. The obtained composites were characterized through FT-IR, XRD, TEM, XPS, and N2 adsorption/desorption isotherms, and the immobilized Cu+ was proven to be active sites. The effects of Cu content, g-C3N4 content, H2O2 concentration, and pH on Rh B degradation were systematically investigated. The effect of the catalyst dose was confirmed with a specific reaction rate constant of (5.9 ± 0.07) × 10−9 m·s−1 and the activation energy was calculated to be 71.0 kJ/mol. In 100 min 96.4% of Rh B (initial concentration 20 mg/L, unadjusted pH (4.9)) was removed in the presence of 1 g/L of catalyst and 10 mM of H2O2 at 25 °C, with an observed reaction rate constant of 6.47 × 10−4 s−1. High degradation rates are achieved at neutral and alkaline conditions and a low copper leaching (0.55 mg/L) was observed even after four reaction cycles. Hydroxyl radical (HO·) was identified as the reactive oxygen species by using isopropanol as a radical scavenger and by ESR analysis. HPLC-MS revealed that the degradation of Rh B on Cu/Al2O3/CN composite involves N-de-ethylation, hydroxylation, de-carboxylation, chromophore cleavage, ring opening, and the mineralization process. Based on the results above, a tentative mechanism for the catalytic performance of the Cu/Al2O3/g-C3N4 composite was proposed. In summary, the characteristics of high degradation rate constants, low ion leaching, and the excellent applicability in neutral and alkaline conditions prove the Cu/Al2O3/g-C3N4 composite to be a superior Fenton-like catalyst compared to many conventional ones.