This paper tackles the challenge of one-bit off-grid direction of arrival (DOA) estimation in a single snapshot scenario based on a learning-based Bayesian approach. Firstly, we formulate the off-grid DOA estimation model, utilizing the first-order off-grid approximation, incorporating one-bit data quantization. Subsequently, we address this problem using the Sparse Bayesian based framework and solve iteratively. However, traditional Sparse Bayesian methods often face challenges such as high computational complexity and the need for extensive hyperparameter tuning. To balance estimation accuracy and computational efficiency, we propose a novel Learning-based Sparse Bayesian framework, which leverages an unrolled neural network architecture. This framework autonomously learns hyperparameters through supervised learning, offering more accurate off-grid DOA estimates and improved computational efficiency compared to some state-of-the-art methods. Furthermore, the proposed approach is applicable to both uniform linear arrays and non-uniform sparse arrays. Simulation results validate the effectiveness of the proposed framework.