We report Eu-local-spin magnetism and Ni-doping-induced superconductivity (SC) in a 112-type ferroarsenide system Eu(Fe$_{1-x}$Ni$_{x}$)As$_2$. The non-doped EuFeAs$_2$ exhibits two primary magnetic transitions at $sim$100 and $sim$ 40 K, probably associated with a spin-density-wave (SDW) transition and an antiferromagnetic ordering in the Fe and Eu sublattices, respectively. Two additional successive transitions possibly related to Eu-spin modulations appear at 15.5 and 6.5 K. For the Ni-doped sample with $x$ = 0.04, the SDW transition disappears, and SC emerges at $T_mathrm{c}$ = 17.5 K. The Eu-spin ordering remains at around 40 K, followed by the possible reentrant magnetic modulations with enhanced spin canting. Consequently, SC coexists with a weak spontaneous magnetization below 6.2 K in Eu(Fe$_{0.96}$Ni$_{0.04}$)As$_2$, which provides a complementary playground for the study of the interplay between SC and magnetism.