In-orbit computing in Low Earth Orbit (LEO) satellite constellations represents a significant advancement in enhancing the efficiency of satellite data processing. Container-based cloud-native solutions are increasingly applied to enhance the elasticity of in-orbit computing. Albeit with high potential, its performance is significantly constrained by the container image downloading delay via satellite-ground links. Therefore, in-orbit container registry is required so as to reduce expensive image downloading overhead. However, due to the motion of LEO satellites, the network topology changes dynamically, leading to fluctuation in the Inter-Satellite Link (ISL) connectivity and communication rates. This imposes significant challenges to in-orbit container registry planning. Additionally, the planning must also account for the limited on-satellite storage and request popularity. To this end, we investigate the problem of in-orbit container registry planning for overall downloading time minimization. The problem is formulated into an ILP form and proved to be NP-hard. We further propose the In-orbit Registry Planning algorithm based on Randomized Rounding (RR-IRP). The experimental results demonstrate the effectiveness of our RR-IRP algorithm, which averagely reduces container image download time by \(22.71\%\) compared to classic solutions.