This work presents prospects for conducting a novel direct test of time-reversal symmetry at the KLOE-2 experiment. Quantum entanglement of neutral $K$ meson pairs uniquely available at KLOE-2 allows to probe directly the time-reversal symmetry ($\mathcal{T}$) independently of $\mathcal{CP}$ violation. This is achieved by a comparison of probabilities for a transition between flavour and $\mathcal{CP}$-definite states and its inverse obtained through exchange of initial and final states. As such, a test requires the reconstruction of the $K_{\rm L}\to 3\pi^0$ decay accompanied by $K_{\rm S}\to \pi^{\pm}\ell^{\mp}\nu$ with good timing information, a new reconstruction method for this process is also presented which is capable of reconstructing the $K_{\rm L}\to 3\pi^0$ decay with decay time resolution of $\mathcal{O}(1\tau_{\rm S})$.