The photoionization-excitation spectrum of Ne i, in which an electron in the $1s$ shell is ionized and the $2p$ orbital electron simultaneously excited to the $np$ orbital, is investigated in detail for complex satellite states ${\mathrm{Ne}}^{+}$ $1{s}^{\text{--}1}2{p}^{\text{--}1}{(}^{1,3}P)np$ $(n=3,4,5,6,7)$. Then, the multiple Auger decay process is studied using the multistep method, including cascade, knock-out, and shake-off mechanisms, based on perturbation theory for the satellite states ${\mathrm{Ne}}^{+}$ $1{s}^{\text{--}1}2{p}^{\text{--}1}({}^{1,3}\mathrm{P})np$ $(n=3,4)$ with two holes. The major single Auger decay can be described as a spectator process that valence electrons remain in the original orbitals. For double Auger decay, direct and cascade processes are considered, in which the direct mechanism is dominated by the spectator process, while the cascade mechanism is mainly from the participator process. For triple Auger decay, the important contribution is from the process consisting of direct double Auger and the subsequent single Auger electron emissions. The calculated ion yields of ${\mathrm{Ne}}^{2+}$, ${\mathrm{Ne}}^{3+}$, and ${\mathrm{Ne}}^{4+}$ for multiple Auger process of satellite states ${\mathrm{Ne}}^{+}$ $1{s}^{\text{--}1}2{p}^{\text{--}1}{(}^{1,3}P)np$ $(n=3,4)$ are consistent with experimental values [Hikosaka et al., Phys. Rev. A 97, 023405 (2018)].