Can you write (with and ink pen) on hydrophobic paper?
This depends on what your ink is based on. If it can be washed off ordinary paper using water, it will be repelled on hydrophobic paper. However, if it can't, it has a chance of staying on.

Note that the paper in particular might repel all or most liquids, not just water. In this case, the ink has a very low chance of staying on the paper.