Instant delivery services, such as food delivery and package delivery, have achieved explosive growth in recent years by providing customers with daily-life convenience. An emerging research area within these services is service Route\&Time Prediction (RTP), which aims to estimate the future service route as well as the arrival time of a given worker. As one of the most crucial tasks in those service platforms, RTP stands central to enhancing user satisfaction and trimming operational expenditures on these platforms. Despite a plethora of algorithms developed to date, there is no systematic, comprehensive survey to guide researchers in this domain. To fill this gap, our work presents the first comprehensive survey that methodically categorizes recent advances in service route and time prediction. We start by defining the RTP challenge and then delve into the metrics that are often employed. Following that, we scrutinize the existing RTP methodologies, presenting a novel taxonomy of them. We categorize these methods based on three criteria: (i) type of task, subdivided into only-route prediction, only-time prediction, and joint route\&time prediction; (ii) model architecture, which encompasses sequence-based and graph-based models; and (iii) learning paradigm, including Supervised Learning (SL) and Deep Reinforcement Learning (DRL). Conclusively, we highlight the limitations of current research and suggest prospective avenues. We believe that the taxonomy, progress, and prospects introduced in this paper can significantly promote the development of this field.