We consider a scheduling problem in a Mobile Edge Caching (MEC) network, where a base station (BS) uploads messages from multiple source nodes (SNs) and transmits them to mobile users (MUs) via downlinks, aiming to jointly optimize the average service Age of Information (AoI) and service delay over MUs. This problem is formulated as a difficult sequential decision making problem with discrete-valued and linearly-constrained design variables. To solve this problem, we first approximate its achievable region by characterizing its superset and subset. The superset is derived based on the rate stability theorem, while the subset is obtained using a novel stochastic policy. We also validate that this subset is substantially identical to the achievable region when the number of schedule resources is large. Additionally, we propose a sufficient condition to check the existence of the solution to the problem. Then, we propose the mixed-order drift-plus-penalty algorithm that uses a dynamic programming (DP) method to optimize the summation over a linear and quadratic Lyapunov drift and a penalty term, to handle the product term over different queue backlogs in the objective function. Finally, by associating the proposed algorithm with the stochastic policy, we demonstrate that it achieves an $O(1/V)$ versus $O(V)$ tradeoff for the average AoI and average delay.