Arthur managed to blend in among the knights in the palace, while also avoiding being seen by those who possibly might recognize him. For instance, he avoided any total chance of passing by Andrew's courtyard.
The library was indeed a secret library just from reading the directions Anna had drawn on the sheet.
Arthur had to navigate through a maze of winding corridors and climb one long spiral staircase that annoyed the hell out of him. The staircase had again led to another corridor, except this corridor looked rather strange. It was so dark that it almost felt as if he had landed in an abyss of darkness, except his feet stood firm on the ground, which proved he wasn't in an abyss.