The detection of early signs of volcanic unrest preceding an eruption, in the form of ground deformation in Interferometric Synthetic Aperture Radar (InSAR) data is critical for assessing volcanic hazard. In this work we treat this as a binary classification problem of InSAR images, and propose a novel deep learning methodology that exploits a rich source of synthetically generated interferograms to train quality classifiers that perform equally well in real interferograms. The imbalanced nature of the problem, with orders of magnitude fewer positive samples, coupled with the lack of a curated database with labeled InSAR data, sets a challenging task for conventional deep learning architectures. We propose a new framework for domain adaptation, in which we learn class prototypes from synthetic data with vision transformers. We report detection accuracy that surpasses the state of the art on volcanic unrest detection. Moreover, we built upon this knowledge by learning a new, non-linear, projection between the learnt representations and prototype space, using pseudo labels produced by our model from an unlabeled real InSAR dataset. This leads to the new state of the art with $97.1%$ accuracy on our test set. We demonstrate the robustness of our approach by training a simple ResNet-18 Convolutional Neural Network on the unlabeled real InSAR dataset with pseudo-labels generated from our top transformer-prototype model. Our methodology provides a significant improvement in performance without the need of manually labeling any sample, opening the road for further exploitation of synthetic InSAR data in various remote sensing applications.