The Internet of Things (IoT) enables the transport of small amounts of data from/to constrained devices such as sensors and actuators. When these devices are located in remote areas without terrestrial network coverage, the Direct-to-Satellite IoT (DtS-IoT) systems become cost-effective solutions. DtS-IoT allows data exchange with remote nodes through low Earth orbit nanosatellites, which are reachable only during short intervals, even during passes with high elevation angles. Therefore, if the constrained nodes become active late concerning a visibility interval, they may miss a transmission opportunity. In the same way, if the nodes become active prematurely, they may incur unnecessary energy consumption. Knowing the duration and start of each visibility interval beforehand allows the constrained devices to increase the probability of successful transmissions and save energy simultaneously. The precise prediction of satellite visibility intervals requires sizable energy and computational costs, a significant challenge for constrained devices, particularly for devices devoid of Internet access on the ground. In this paper, we propose a fast and efficient algorithm for satellite visibility prediction that considers the two-body model as a reference and includes a mechanism for the constrained ground nodes to update the orbital information. Numerical and simulation results show that our solution decreases the computation cost by up to 66.53% compared to state-of-the-art visibility prediction algorithms. Furthermore, we show that for each pass of the satellite, the constrained devices successfully update the orbital information 87.12% of the time, this process being completely independent of any supplementary Internet connectivity.