Underground engineering, especially deep geotechnical engineering, is often affected alternately by groundwater infiltration and ventilation drying during construction and use. In addition, acid rain, mineral dissolution, acid deposition, and other factors make groundwater acidic. This caused the underground structure to erode and has also threatened its safety and durability. In this study, both the physical and mechanical properties under acid dry-wet (A-D-W) cycles were investigated. Deep sandstone was treated repeatedly under acid cycling, and its physical parameters were measured. Uniaxial compressive strength tests and microtests were carried out. Finally, using a combination of scanning electron microscopy (SEM) and backscatter electron images (BSE), the microstructural changes of sandstone under the combined action of an acidic environment and a dry-wet cycle were described. The test results show that the mass, P-wave velocity, peak stress, and elastic modulus of sandstone after the A-D-W cycles decreased by 0.43%, 7.87%, 70.20%, and 88.10%, respectively. With the increase in the number of cycles, the loss of these indicators increased. However, the peak strain increased with an average increase of 55.8%. In addition, 10 cycles are the critical point for physical and mechanical indicators of the A-D-W cycles. After 10 cycles, the changes of various indicators increase rapidly. Microscopic analysis shows that the reasons of this phenomenon were that the specimen was corroded by the sulfuric acid solution, which resulted in the development of pores and cracks and the decline of physical and mechanical properties. Conversely, the development of pores was hindered by sediment, which slowed down the decline rate of the physical and mechanical indexes of specimens, but this phenomenon disappeared with the increase of A-D-W cycles. Based on the analysis of the experimental phenomena, the constitutive model of uniaxial compression based on Weibull damage variable has been established in this article, and the model has the best effect in acid solution with less cycle times or pH value of solution greater than 6.