From experimental and theoretical analyses of magnetic and specific-heat properties, we present the complete magnetic phase diagram of the quasi-one-dimensional antiferromagnet Cu(N$_2$H$_5$)$_2$(SO$_4$)$_2$. On cooling and at zero magnetic field this compound enters a one-dimensional regime with short-range magnetic correlations, marked by a broad maximum in the specific heat and magnetic susceptibility at $T_\mathrm{max}\sim 2\,\mathrm{K}$, followed by an ordered antiferromagnetic phase below $T_\mathrm{N}\sim 1\,\mathrm{K}$ induced by small interchain couplings... The intermediate-temperate one-dimensional regime can be modeled using exact quantum-transfer-matrix calculations, which perfectly describe the nonmonotonic behavior of T_max as a function of the magnetic field, giving $J = 4.25\,\mathrm{K}$ for the intrachain exchange parameter. The analysis of magnetic specific-heat and susceptibility data at low temperature indicates that the interchain exchange couplings are an order of magnitude smaller than the coupling inside the chains. (read more)