We consider a service cost minimization problem for resource-constrained small-cell networks with caching, where the challenge mainly stems from (i) the insufficient backhaul capacity and limited network bandwidth and (ii) the limited storing capacity of small-cell base stations (SBSs). Besides, the optimization problem is NP-hard since both the users' mobility patterns and content preferences are unknown. In this paper, we develop a novel mobility-aware joint routing and caching strategy to address the challenges. The designed framework divides the entire geographical area into small sections containing one SBS and several MUs. Based on the concept of one-stop-shop (OSS), we propose a federated routing and popularity learning (FRPL) approach in which the SBSs cooperatively learn the routing and preference of their respective MUs and make a caching decision. The FRPL method completes multiple tasks in one shot, thus reducing the average processing time per global aggregation of learning. By exploiting the outcomes of FRPL together with the estimated service edge of SBSs, the proposed cache placement solution greedily approximates the minimizer of the challenging service cost optimization problem. Theoretical and numerical analyses show the effectiveness of our proposed approaches.