The isomerization reaction between two discernible categories of structures of Mg+Ar12 with the magnesium cation inside the argon cluster (solvated), and on the surface of the argon cluster (non-solvated), discovered recently, is studied by computing the density of stales with microcanonical and canonical molecular dynamics techniques. Thermodynamical properties and equilibrium constants are then evaluated after establishing the validity of a single effective transition state. A complex kinetic model based on the transition state theory and the harmonic approximation is also developed to calculate thermodynamical properties, and isomerization rate and equilibrium constants. This model includes tens of intermediate minima and transition states located on the potential energy surface of the cluster. The general good agreement between the two models is encouraging for studying complex isomerization processes in atomic clusters with simplified theories.