Is it permissible to travel from Saudi Arabia to Egypt for one day to take an exam for a government job, without the employer's knowledge, and only with the direct manager's knowledge, for fear of refusal?

It is permissible to be absent with the permission of your direct superior in Saudi Arabia, if he has the authority to grant permission for absence due to need, provided that the permission is not a favoritism towards you. If he does not have the authority, then the permission of the authorized person is necessary. You must continue working until the end of the contract, based on the Almighty's saying: "O you who have believed, fulfill [all] contracts," and the Prophet's (peace be upon him) saying: "Muslims are bound by their conditions in what is lawful."