Groundwater recharge estimation in arid regions is challenged by data scarcity and high uncertainty. This study presents a Physics-Informed Bayesian Neural Network (PI-BNN) to quantify groundwater recharge and its uncertainty in the South Al Batinah (SAB) Basin, northern Oman. The PI-BNN was applied within a soil-moisture mass balance framework using 34 years (1990-2023) of monthly FLDAS remote sensing data, embedding the water balance equation directly into the training loss function through nine physics-informed penalty terms. Results were compared against Latin Hypercube Sampling (LHS) using the same inputs. Recharge is highly seasonal and episodic, peaking in December (similar to 5-6 mm/month) with moderate values in February-March and July, and negligible recharge during dry months. Annual recharge is estimated at approximately 16 mm/yr (PI-BNN) and 7-11 mm/yr (LHS). Precipitation variability accounts for more than 70% of recharge uncertainty during wet months. The PI-BNN reduces uncertainty bounds by approximately 50% compared to independent LHS while maintaining physically consistent estimates. An ablation experiment confirms that the physics-informed constraints, rather than the Bayesian architecture alone, drive physically plausible recharge recovery. The proposed methodology offers a robust and transferable framework for recharge estimation in data-scarce arid environments.