Intracranial EEG (iEEG) provides high-fidelity neural recordings essential for clinical and brain-computer interface applications, but acquiring these signals requires invasive surgery. While recent studies have attempted to estimate iEEG from non-invasive scalp EEG, most rely on patient-specific models, creating a circular dependency: if surgery is required to collect training data, the non-invasive model offers limited practical benefit. In this study, we address the challenge of cross-subject iEEG reconstruction by predicting intracranial signals for unseen patients using models trained on other individuals. We propose CAST (Cross-Attention Spatial-Temporal Transformer), a machine learning framework that translates scalp EEG into multi-channel iEEG waveforms through a two-stage transfer learning strategy. First, a temporal encoder extracts multi-scale neural representations at three different resolutions. Then, because electrode placements vary substantially across patients, a channel-aware decoder is calibrated using only a few minutes of data from the target subject. We evaluated the proposed method using leave-one-subject-out cross-validation on two public datasets comprising 1,282 iEEG channels. Experimental results demonstrate that CAST reconstructs cortical signals located near the scalp surface substantially better than deep subcortical activity. In highly observable sensorimotor regions, the model achieved peak correlations of up to r=0.864 in the precentral gyrus. Furthermore, with a channel selection strategy, CAST obtained a mean correlation of r=0.545 on viable subjects, outperforming previous within-subject baselines. These findings indicate that cortical iEEG signals can be reconstructed for unseen subjects from scalp EEG without extensive patient-specific training, and that only a brief calibration phase is sufficient to adapt the model to new hardware configurations.