The effect of Cu substitution on the structural and superconducting properties of the $(Ru_{1-x}Cu_x)Sr_2(Eu_{1.34{cdot}}Ce_{0.66})Cu_2O_z$ system with x = 0, 0.25 and 0.5 prepared under ambient pressure have been investigated. The X-ray diffraction patterns indicated that the Ru ions are replaced by the Cu ions. It is found that the Cu substitution for Ru significantly reduces the ferromagnetic component of field-cooled magnetic susceptibility, but results in a small change in diamagnetic onset transition temperature of zero-field-cooled magnetic susceptibility. In contrast to the Ru $Sr_2(Eu_{1.34{cdot}}Ce_{0.66})Cu_2O_z$, bulk Meissner effect is observed in the field-cooled magnetization measurements of the Cu doped samples. The experimental results are discussed in connection with the spontaneous vortex phase interpretation.