We report a method to determine time evolution of ortho-positronium ($o$-Ps) kinetic energy with higher accuracy and sensitivity than conventional methods. Our method utilizes the fact that, during Ps-Xe collisions, an $o$-Ps atom can undergo a spin-conversion reaction induced by spin-orbit interaction to decay into two gamma-ray photons and the fact that the reaction rate strongly depends on the $o$-Ps kinetic energy because the reaction occurs only in $p$-wave scattering; thus a small change in the energy leads to a large change in the two-photon annihilation rate. Utilizing this reaction as a ``lens'' to magnify the $o$-Ps kinetic energy, we obtain its time evolution by measuring the time-resolved two-photon annihilation rate with an age-momentum correlation spectrometer. The time evolution of $o$-Ps kinetic energy can be explained by a classical model that assumes elastic collisions in a time range greater than 20 ns and an energy range of 40--60 meV. By applying this method, the Ps-Xe momentum-transfer cross section is found to be $12(2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}16}$ cm${}^{2}$.