Getting permission to leave the school and the mountain domain wasn't too hard, not after all the years he's spent proving he could handle things on his own.
There was not much that Gakusha had feared for if it came to obstacles that'd appear before Seiya's path and while he was still young, his newfound experience would make up for the gap in age between him and whoever else he might possibly meet that may pose a danger to the prince.
This was of course dependent on whether or not Seiya didn't rattle their nerves and made them flee instead of seeking conflict.
Seiya's journey to the Palace was a quick venture, especially since he learned how to increase his movement through a technique he had learned while in Gakusha's school, a skill that allowed him to traverse great distances in a swift action although it was far more taxing with greater distances naturally but at least it allowed him to reach his destination in such a farther window of time as it would take him before.