As a promising key technology of 6th-Generation (6G) mobile communication systems, integrated sensing and communication (ISAC) technology aims to make full use of spectrum resources to enable the functional integration of communication and sensing. The ISAC-enabled mobile communication systems regularly operate in non-continuous spectrum bands due to crowded licensed frequency bands. However, the conventional sensing algorithms over non-continuous spectrum bands have disadvantages such as reduced peak-to-sidelobe ratio (PSR) and degraded anti-noise performance. Facing this challenge, we propose a high-precision ISAC signal processing algorithm based on compressed sensing (CS) in this paper. By integrating the resource block group (RBG) configuration information in 5th-Generation new radio (5G NR) and channel information matrices, we can dynamically and accurately obtain power estimation spectra. Moreover, we employ the fast iterative shrinkage-thresholding algorithm (FISTA) to address the reconstruction problem and utilize K-fold cross validation (KCV) to obtain optimal parameters. Simulation results show that the proposed algorithm has lower sidelobes or even zero sidelobes and high anti-noise performance compared with conventional sensing algorithms.