An appropriations law granting the legislators lump-sum funds in which they have full discretion on what project it would fund and how much the project would cost, was passed. Is such law constitutional?

No, such law is unconstitutional. The law violates the Principle of Separation of Powers when it allowed legislators to wield budget execution. It also violates the Principle of non-delegation of legislative power when it conferred unto legislators the power of appropriation by giving them discretion to fund specific projects which they themselves determine.