Accurate prediction of the remaining useful life (RUL) of industrial machinery is essential for reducing downtime and optimizing maintenance schedules. Existing approaches, such as long short-term memory (LSTM) networks and convolutional neural networks (CNNs), often struggle to model both global temporal dependencies and fine-grained degradation trends in multivariate sensor data. We propose a hybrid model, FTT-GRU, which combines a Fast Temporal Transformer (FTT) -- a lightweight Transformer variant using linearized attention via fast Fourier transform (FFT) -- with a gated recurrent unit (GRU) layer for sequential modeling. To the best of our knowledge, this is the first application of an FTT with a GRU for RUL prediction on NASA CMAPSS, enabling simultaneous capture of global and local degradation patterns in a compact architecture. On CMAPSS FD001, FTT-GRU attains RMSE 30.76, MAE 18.97, and $R^2=0.45$, with 1.12 ms CPU latency at batch=1. Relative to the best published deep baseline (TCN--Attention), it improves RMSE by 1.16\% and MAE by 4.00\%. Training curves averaged over $k=3$ runs show smooth convergence with narrow 95\% confidence bands, and ablations (GRU-only, FTT-only) support the contribution of both components. These results demonstrate that a compact Transformer-RNN hybrid delivers accurate and efficient RUL predictions on CMAPSS, making it suitable for real-time industrial prognostics.