Mobile Edge Caching (MEC) integrated with Deep Neural Networks (DNNs) is an innovative technology with significant potential for the future generation of wireless networks, resulting in a considerable reduction in users' latency. The MEC network's effectiveness, however, heavily relies on its capacity to predict and dynamically update the storage of caching nodes with the most popular contents. To be effective, a DNN-based popularity prediction model needs to have the ability to understand the historical request patterns of content, including their temporal and spatial correlations. Existing state-of-the-art time-series DNN models capture the latter by simultaneously inputting the sequential request patterns of multiple contents to the network, considerably increasing the size of the input sample. This motivates us to address this challenge by proposing a DNN-based popularity prediction framework based on the idea of contrasting input samples against each other, designed for the Unmanned Aerial Vehicle (UAV)-aided MEC networks. Referred to as the Contrastive Learning-based Survival Analysis (CLSA), the proposed architecture consists of a self-supervised Contrastive Learning (CL) model, where the temporal information of sequential requests is learned using a Long Short Term Memory (LSTM) network as the encoder of the CL architecture. Followed by a Survival Analysis (SA) network, the output of the proposed CLSA architecture is probabilities for each content's future popularity, which are then sorted in descending order to identify the Top-K popular contents. Based on the simulation results, the proposed CLSA architecture outperforms its counterparts across the classification accuracy and cache-hit ratio.