Reliable navigation in urban environments remains challenging due to non-Gaussian noise and time-varying error characteristics that degrade the performance of integrated Global Navigation Satellite System/Inertial Navigation System (GNSS/INS) solutions. This paper proposes a novel adaptive filtering method based on a Shallow Neural Network (SNN) incorporated into a Maximum Correntropy Unscented Kalman Filter (MCUKF) framework. The proposed SNN-MCUKF dynamically adjusts the robustness factor, known as the kernel bandwidth, by learning from the statistical history of measurement residuals. This adaptive mechanism enables the filter to maintain both robustness and statistical consistency even under rapidly changing environmental conditions. To achieve this, the SNN estimates the kernel bandwidth online using a self-supervised learning approach driven by the exponential moving average of the residual energy. This allows the filter to respond in real time to variations in measurement errors such as multipath and non-line-of-sight effects, which are common in dense urban areas. Experiments were conducted using a GNSS/INS platform operating in mixed urban environments with varying levels of signal obstruction. The results demonstrate that the proposed method achieves significantly improved estimation stability compared with conventional Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), and fixed-bandwidth MCC-based filters. Notably, the SNN-MCUKF effectively suppresses long-term drift in attitude estimation and maintains consistent position and velocity accuracy even under severe multipath interference. These findings confirm that the SNN-MCUKF provides a practical and computationally efficient solution for integrated navigation in non-Gaussian and dynamic conditions, enhancing both the reliability and robustness of GNSS/INS fusion systems.