Modern astrophysics and cosmology increasingly rely on simulations and cross-survey analyses, yet differences in instrumentation, calibration, and modeling introduce distribution mismatches between simulated and observed datasets (domain shift). This occurs when the joint distribution of inputs and labels differs between the training (source) and target domains, so source-trained models underperform on the target. Domain adaptation (DA), the branch of transfer learning for this case---fixed task, shifted input--label distribution---provides principled ways to mitigate it. We study a concrete simulation-to-observation (sim-to-obs) case: semisupervised domain adaptation (SSDA) to adapt a four-class spectral classifier (high-redshift quasars, low-redshift quasars, galaxies, and stars) from DESI-based J-PAS mock catalogs to real J-PAS photometry. Because labels are plentiful in simulations but scarce in the target survey, our pipeline pretrains on labeled DESI-->J-PAS mocks and adapts using a small, labeled J-PAS subset (of which only 242 are high-redshift quasars). We benchmark SSDA against a J-PAS--only-supervised model (same labeled subset) and a mock-only model-evaluated zero-shot on J-PAS, plus the same model on its in-domain mock test set (four evaluations total). On held-out J-PAS data, SSDA achieves a macro-F1 of 0.83 and a true positive rate of 0.90, versus 0.79/0.85 and 0.73/0.87 for the supervised and mock-only baselines. The gains come mainly from improved quasar classification, especially the high-redshift subclass (F1 = 0.68, versus 0.54 and 0.40 for those baselines), yielding better-calibrated candidate lists for spectroscopic targeting (e.g., WEAVE-QSO) and active galactic nucleus searches. This shows how a few labeled target examples enable robust, data-efficient transfer.