Simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) has emerged as a promising technology to realize full-space coverage and boost spectral efficiency in next-generation wireless networks. Yet, the joint design of the base station precoding matrix as well as the STAR-RIS transmission and reflection coefficient matrices leads to a high-dimensional, strongly nonconvex, and NP-hard optimization problem. Conventional alternating optimization (AO) schemes typically involve repeated large-scale matrix inversion operations, resulting in high computational complexity and poor scalability, while existing deep learning approaches often rely on expensive pre-training and large network models. In this paper, we develop a gradient-based meta learning (GML) framework that directly feeds optimization gradients into lightweight neural networks, thereby removing the need for pre-training and enabling fast adaptation. Specifically, we design dedicated GML-based schemes for both independent-phase and coupled-phase STAR-RIS models, effectively handling their respective amplitude and phase constraints while achieving weighted sum-rate performance very close to that of AO-based benchmarks. Extensive simulations demonstrate that, for both phase models, the proposed methods substantially reduce computational overhead, with complexity growing nearly linearly when the number of BS antennas and STAR-RIS elements grows, and yielding up to 10 times runtime speedup over AO, which confirms the scalability and practicality of the proposed GML method for large-scale STAR-RIS-assisted communications.