Neutron tomography is an efficient nondestructive testing technique. As a complement to X-ray computed tomography, it has been widely used in various fields. Due to the difficulty of obtaining complete neutron projection data in a high-radiation environment and the high noise characteristics of neutron images, it is difficult to reconstruct a high-quality image using the conventional filtered-back projection (FBP) algorithm. Therefore, research on sparse-view reconstruction algorithms in neutron tomography is needed. To improve the quality of neutron three-dimensional reconstructed images, this paper proposes an algorithm that combines the Simultaneous Algebraic Reconstruction Technique (SART) with Fast Gradient Projection (FGP), where the FGP is an algorithm for image denoising and deblurring based on the discrete total variation (TV) minimization model. The algorithm proposed in this paper is compared with other algorithms (FBP, SART, and SART-TV) by simulated experimental data and real neutron experimental data. The experimental results show that the novel algorithm outperforms the other three algorithms in terms of denoising and retaining detailed structural information.