Is it permissible to accept a fee from someone funding the construction of rooms for supervising the building work, knowing that this task is additional and not part of his primary duties as an assistant manager, and that the supervisor is usually paid $500 for it?

It is not permissible to take anything from the building fund entrusted to you, as this constitutes deception and a betrayal of trust. Allah has commanded us to fulfill trusts and abide by contracts. You can refuse to do additional work until you are compensated for it. However, performing work under the guise of volunteering and then secretly taking payment for it is a betrayal. Perfecting one's work for the sake of Allah is better than the transient things of this world.