Coded aperture snapshot spectral imaging is a computational imaging technique used to reconstruct three-dimensional hyperspectral images from one or several two-dimensional projection measurements. However, fewer projection measurements or more spectral channels lead to a severely ill-posed problem, in which case regularization methods have to be applied. In order to significantly improve the accuracy of reconstruction, this paper proposes a fast alternating minimization algorithm based on the sparsity and deep image priors of natural images. By integrating deep image prior into the principle of compressive sensing reconstruction, the proposed algorithm can achieve state-of-the-art results without any training dataset. Extensive experiments show that the Fama-SDIP method significantly outperforms prevailing methods on simulation and real HSI datasets.