We use first-principles calculations to study pressure effects on the vibrational and superconducting properties of H$_3$S in the cubic $Im\bar{3}m$ phase for the pressure range where the superconducting critical temperature (T$_c$) was measured (155-225 GPa). The pressure effects were incorporated using the Functional Derivative Method (FDM). In this paper, we present for the first time, the Cooper-Pairs Distribution Functions D$_{cp}$($\omega,T_c$) for H$_3$S, which will allow to identify the spectral regions where Cooper-Pairs formation at temperature T$_c$ is more favorable. We analyzed in detail the pressure effects on the electron-phonon spectral density function $\alpha^2$F$(\omega)$ and phonon density of states (PhDOS) and their relationship with T$_c$. The FDM manages to reproduce the trend of the pressure dependence of critical temperature, in good agreement with experimental data in the range of 155 to 190 GPa. The $D_{cp}(\omega,T_c)$ suggests that the low-frequency vibration region is where Cooper-Pairs are possible, which means that S-vibrations have an important role in the H$_3$S superconductivity properties.