Fiber-reinforced alkali-slag recycled concrete, a novel green building material with high-efficiency solid-waste utilization, can effectively utilize waste slag and recover coarse aggregates. However, no relevant research has been conducted on the mechanical properties under axial compression. Therefore, this study investigated the influence of fiber type, fiber content, and recycled coarse aggregate content on the axial compression mechanical properties of fiber-reinforced alkali-slag recycled concrete. The results show that an appropriate amount of steel, basalt, and polypropylene fibers can effectively improve the axial compressive strength, peak compressive strain, and elastic modulus of alkali-activated slag recycled concrete. The axial compressive strength, peak compressive strain, and elasticity can be enhanced by 10.6–31.9%, 26.0–29.3%, and 1.4–69.1%, respectively. Basalt fibers had the worst effect, while steel fibers had the best effect. A small proportion of recycled coarse aggregate can slightly increase the axial compressive strength and peak compressive strain while significantly reducing the elastic modulus. An axial compression stress–strain model of the fiber-reinforced alkali-slag recycled concrete was established. These results provide basic theoretical support for the comprehensive and effective exploitation of waste slag and building waste.