What is the ruling of the Shari'ah on an employee taking money from his brother in exchange for expediting his affairs in the department where he works, and is it permissible for him to take this money without hesitation?

You are permitted to take the amount your brother has designated for handling his affairs, because it falls under the category of a Ju'alah (a stipulated reward for a specific task) which is permissible in Islamic law. The evidence for its permissibility is Allah's saying: "And for him who brings it [there shall be] a camel's load; and I am guarantor for it" (Quran 12:72). This is on condition that what you take is not in exchange for work that you are already obligated to do by virtue of your employment; anything you take for that type of work would be a bribe.