I suspect this is simpler than it appears. The text of Neh 2:1 does NOT say that Nehemiah was afraid of the king, although that may have been part of the reason.
The primary cause of Nehemiah's fear was his overwhelming sense of duty and responsibility to his people and a realization that he had been called of God to do a work for the Jews in Jerusalem. He now has an opportunity to do this and he does not want to miss it.
Thus, I believe Nehemiah was afraid of not being adequate for the job and saying something to king that would spoil his chances of gaining royal approval to help the Jews. Nehemiah' sensed his human inadequacies and wanted to be sensitive to the leading of God.
Benson suggests something similar:
And he feared a disappointment, because his request was great and
invidious, and odious to most of the Persian courtiers.
Barnes has a similar idea:
I was very sore afraid - A Persian subject was expected to be
perfectly content so long as he had the happiness of being with his
king. A request to quit the court was thus a serious matter.
Matthew Poole expresses it this way:
I was very sore afraid; partly, being daunted by the majesty of the king, and the suddenness and sharpness of his question; partly,
fearing lest there was arising some jealousy or ill opinion in the
king concerning him; partly, because it was an unusual and ungrateful
thing to come into the king of Persia’s presence with any badges or
tokens of sorrow, Esther 4:2; and principally, from his doubts or
fears of disappointment, because his request was great and
invidious, and odious to the most of the Persian courtiers, and might
be represented as dangerous, and might seem improper for a time of
feasting and jollity.