There has been an increasing interest in mobile edge computing (MEC) in recent years. Different from the traditional centralized cloud computing, MEC servers are deployed at the edges of networks such as at base stations (BSs) and access points (APs), in order to support computation-intensive and latency-critical applications. In this paper, we consider a multi-user mobile edge computing (MEC) network in which the wireless data transmission/offloading is performed using finite blocklength (FBL) codes to satisfy the latency constraints. The reliability in the communication phase is characterized in the FBL regime, while the event of queue length violation in the computation phase is investigated via exploiting the extreme value theory. We first formulate the overall optimization problem in the scenario of multiple user equipments (UEs) aiming to minimize the maximal end-to-end error probability among all UEs under both the FBL and energy consumption constraints. We further propose a two-level learning-based approach to jointly determine time allocations in the FBL regime, the UEs’ offloading decisions and MEC computational resource allocations to solve the overall optimization problem. Simulation results demonstrate that the proposed two-level learning-based algorithm solves the problem efficiently.