Hyperspectral pansharpening refers to fusing a panchromatic image (PAN) and a low-resolution hyperspectral image (LR-HSI) to obtain a high-resolution hyperspectral image (HR-HSI). Recently, guiding pre-trained diffusion models (DMs) has demonstrated significant potential in this area, leveraging their powerful representational abilities while avoiding complex training processes. However, these DMs are often trained on RGB images, not well-suited for pansharpening tasks, limited in adapting to the hyperspectral images. In this work, we propose a novel guided diffusion scheme with zero-shot guidance and neural spatial-spectral decomposition (NSSD) to iteratively generate the RGB detail image and map the RGB detail image to target HR-HSI. Specifically, zero-shot guidance employs an auxiliary neural network that trained only with a PAN and LR-HSI to guide pre-trained DMs in generating the RGB detail image, informed by specific prior knowledge. Then, NSSD establishes a spectral mapping from the generated RGB detail image to the final HR-HSI. Extensive experiments are conducted on Pavia, Washington DC, and Chukusei datasets to demonstrate that the proposed method significantly enhances the performance of DMs for hyperspectral pansharpening tasks, outperforming existing methods across multiple metrics and achieving improvements in visualization results.