The coded aperture snapshot spectral imaging system, based on compressed sensing theory, functions as an capable of efficiently acquiring compressed two-dimensional spectral data. This data is subsequently decoded into three-dimensional spectral data through a deep neural network. However, training the deep neural network necessitates a substantial amount of clean data, which is often challenging to obtain. To address the issue of insufficient training data for deep neural network, a self-supervised hyperspectral denoising neural network is proposed, leveraging the concept of neighborhood sampling. This network is integrated into the deep plug-and-play framework, enabling self-supervised spectral reconstruction. The study also examines the impact of different noise degradation models on the final reconstruction quality. Experimental results demonstrate that compared with supervised learning method, the self-supervised learning method enhances the average peak signal-to-noise ratio by 1.18dB and improves the structural similarity is improved by 0.009. Additionally, it achieves superior visual reconstruction outcomes without relying on clean data as labels.