You can apply the same principal component transformation from the source dataset to the target dataset. This will map your target dataset from the initial N dimensions to the same M dimensions, putting the source and target data in the same PCA space. To label the new points, you can train a classifier on the source data, using any supervised method to identify the discovered classes from PCA features (or the original features, if you prefer). Finally, apply the classifier to the target data to classify each target point as one of the clusters you discovered. Note that traditional classifier metrics such as ROC or accuracy are not terribly meaningful in this case, since there is no independent test set, as all the samples were included in the clustering. A supervised method will basically always do a good job of identifying classes that were found through unsupervised clustering on the same data in the first place.
Alternatively, you could perform a single clustering with both the source and target data combined, although you will likely get somewhat different results than just clustering the source data alone, which may not be what you want to do.