Extended Kalman Filter (EKF) is extensively employed in Global Navigation Satellite System (GNSS)-based real-time orbit determination (RTOD) for microsatellites due to its low complexity. However, the performance of EKF-RTOD is markedly degraded when the microsatellite deviates from a stable Earth-pointing attitude and employs a low-cost receiver. Factor graph optimization (FGO), which addresses nonlinear problems through multiple iterations and re-linearization, has demonstrated superior accuracy and robustness compared to EKF in challenging environments such as urban canyons. In this study, we introduce a novel FGO-based RTOD (FGO-RTOD) approach, which integrates state transfer factors to establish temporal connections between state nodes across multiple epochs. Real-time processing is achieved through a sliding window mechanism combined with marginalization. This paper evaluates the performance of the proposed algorithm in a regular scenario using data from GRACE-FO-A, which maintains the Earth-pointing attitude and employs a high-performance receiver. The positioning results of GRACE-FO-A indicate that FGO-RTOD marginally outperforms EKF-RTOD in accuracy. Furthermore, the performance of FGO-RTOD is assessed in challenging scenarios using simulation data and on-orbit data from Tianping-2B microsatellite, which is not in an Earth-pointing attitude and employs a low-cost receiver. The simulation results reveal that FGO-RTOD reduces the Root Mean Square (RMS) of positioning error by 79.0% relative to EKF-RTOD and exhibits significantly enhanced smoothing. In the Tianping-2B experiments, FGO-RTOD reduces the RMS of carrier-phase ionosphere-free combination residuals from 2 cm to 1 cm relative to EKF-RTOD, alongside a substantial improvement in the ratio of valid observations. These findings underscore the effectiveness of FGO-RTOD in managing outlier measurements in challenging scenarios.