Low Earth Orbit (LEO) satellite mega-constellations such as Starlink require frequent handovers due to high orbital velocities, resulting in substantial Round-Trip Time (RTT) fluctuations. Accurate RTT prediction can mitigate these fluctuations, yet it fundamentally relies on continuously updated serving- satellite information. Existing serving-satellite identification methods typically complete identification only at the end of each scheduling cycle, creating an identification blind zone that limits real-time prediction. To overcome this limitation, we develop StarRTT, an end-to-end real-time framework that eliminates the blind zone by integrating per-second serving-satellite identification with RTT prediction. Within this framework, we propose Real-Time Long Short-Term Memory (RT-LSTM), an LSTM-based RTT prediction model designed to capture both non-stationary RTT evolution and time-varying geometric relationships among the serving satellite, user terminal, and gateways. Specifically, a historical LSTM encodes historical RTT observations and real-time serving-satellite and gateway-relative features, while a future LSTM processes Two-Line Element (TLE)-propagated future features of the identified serving satellite up to the next scheduling instant. Under strict real-time back-testing, RT-LSTM achieves a Mean Absolute Error (MAE) of 1.405 ms averaged over all 15 prediction horizons and 15 reference time-slot indices, reducing the error by 4.3 %, 11.6 %, and 22.1 % compared with the existing baseline models, LSTM, StarNet, and Attention-based Spatiotemporal Model (ASTM), respectively. Notably, the reduction over LSTM reaches 12.5 % at the scheduling instant, highlighting the benefit of real-time serving-satellite information when recent RTT observations are limited.

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