Proactively caching content at the network edge is particularly effective in high-mobility vehicular networks, where intermittent connection is the major challenge for seamless content transmission. The objective of this paper is to achieve proactive caching in vehicular networks by mobility prediction, specifically by predicting the next roadside unit (RSU) for a vehicle with reinforcement learning techniques. The paper proposes two proactive caching algorithms based on multi-armed bandit (MAB) learning, non-contextual MAB and contextual MAB, respectively. This paper fills the void in the literature regarding the application of MAB learning to mobility-prediction based proactive caching. Their feasibility, superiority, and applicability are evaluated with simulation in two modern cities: Las Vegas, USA with a grid road layout, and Manchester, UK with a more historical layout. Additionally, this paper is the first that proposes to investigate the uncertainty associated with proactive caching systems in the form of entropy with a specifically extended Subjective Logic framework, in order to provide an insight into the underlying link between prediction accuracy and uncertainty.