Is it permissible to change the job title in the passport to enable working in Saudi Arabia, while informing the employer of the real job, and paying a sum of money to a public employee to facilitate that?

One must abide by the laws of the state, so long as they do not contradict Islamic Sharia. Paying money to an employee to change a job title is considered a bribe, and both the briber and the bribed are cursed. However, if the state laws permit changing the job title, and you are in urgent need of this change for work, and it cannot be achieved except by paying a bribe, then it is permissible to pay it, and the sin falls upon the recipient of the bribe. But if any of these conditions are not met, then it is not permissible to do so.