Dumps of waste metallurgical slags can store large quantities of critical raw materials. This study examined the granulated slag from a Ni-Cu smelter, containing 0.30 wt% Cu, 0.11 wt% Ni, and 0.086 wt% Co. A combination of mineralogical characterization techniques (XRD, SEM-EPMA, LA-ICP-MS, and automated mineralogy) was employed to investigate the phase composition and distribution of target elements to provide the key information for the possible reprocessing of this material on a quantitative basis. The major constituents are silicate glass (84 area%) and fayalite (15 area%), while entrapped matte (sulfide inclusions) accounts only for 0.2 area%. The silicate phases are also the primary hosts for the target elements, accounting for 98%, 74%, and 66% of the total Co, Cu, and Ni, respectively. Our results indicate that acid leaching and hydrometallurgical processing are the preferred strategies for the potential metal recovery. Laboratory extraction tests were conducted under the following conditions: original slag without comminution and finely milled; sulfuric acid, methanesulfonic acid (MSA), and citric acid; liquid/solid ratios of 10 and 20 L/kg; ambient and elevated (70 °C) temperatures; 2 and 6 h of leaching. For the original granulated slag, extraction efficiencies up to 89% for Co, 71% for Cu, and 46% for Ni were reached. The extractions of Cu and Ni were limited by adsorption or incorporation into newly formed phases. Considering the current prices of MSA and citric acid and the lack of improvement in the extraction efficiency, their use for metal recovery instead of sulfuric acid appears unlikely.