We address the challenging problem of estimating the directions-of-arrival (DOAs) of multiple off-grid signals using a single snapshot of one-bit quantized measurements. Conventional DOA estimation methods face difficulties in tackling this problem effectively. This paper introduces a domain-knowledge-guided learning framework to achieve high-resolution DOA estimation in such a scenario, thus drastically reducing hardware complexity without compromising performance. We first reformulate DOA estimation as a maximum a posteriori (MAP) problem, unifying on-grid and off-grid scenarios under a Laplacian-type sparsity prior to effectively enforce sparsity for both uniform and sparse linear arrays. For off-grid signals, a first-order approximation grid model is embedded into the one-bit signal model. We then reinterpret one-bit sensing as a binary classification task, employing a multivariate Bernoulli likelihood with a logistic link function to enhance stability and estimation accuracy. To resolve the non-convexity inherent in the MAP formulation, we develop augmented algorithmic frameworks based on majorization-minimization principles. Further, we design model-based inference neural networks by deep unrolling these frameworks, significantly reducing computational complexity while preserving the estimation precision. Extensive simulations demonstrate the robustness of the proposed framework across a wide range of input signal-to-noise ratio values and off-grid deviations. By integrating the unified model-based priors with data-driven learning, this work bridges the gap between theoretical guarantees and practical feasibility in one-bit single-snapshot DOA estimation, offering a scalable, hardware-efficient solution for next-generation radar and communication systems.