With the development of low earth orbits (LEO) high throughput satellite (HTS), satellite-Internet of Things (S-IoT) has become a crucial direction in beyond 5G (B5G) and future sixth generation (6G) mobile system due to the ubiquitous coverage and broadband access capability inherited from the HTS. Several S-IoT applications for monitoring status updates can be seen as classic cases of real time transmission, where the main performance parameters are the information freshness and fairness of the network, i.e., the age of information (AoI) and Jain's Fairness Index (JFI), respectively. In this paper, we propose a freshness-critical incremental-redundancy hybrid automatic repeat request (FCIR-HARQ) multi-hop transmission scheme to improve the AoI and JFI performance, where each hop adopts the Last Come First Serve with preemption only in waiting (LCFS-W) policy with a buffer only store the latest file. In particular, we derive the closed form expression of average AoI (AAoI) in the above multi-hop S-IoT, and analyze the corresponding JFI in each hop. Simulation results validate the accuracy of our derivations, and show that the proposed FCIR-HARQ multi-hop transmission scheme with LCFS- W policy outperforms the existing First Come First Serve (FCFS) policy in terms of AAoI.