Recently “the Hat” monotile was introduced into the family of aperiodic tilings and quasicrystals boasting physical properties lying at the boundary of ordered and disordered systems. Here we study the two-dimensional wave transport, transverse localization and scaling properties of the quantum modes in a Monotile quasilattice. Our system is based on reconfigurable optical lattices for cavity-polaritons which provide flexible means to study wavepacket dynamics, strong nonlinear phenomena, and power-driven condensation in this new type of an aperiodic tiling. We confirm the existence of localized and critical states in the Monotile through direct diagonalization of the Schrödinger equation. Scaling analysis on the moments of the wavefunction distribution reveals anomalous transport regimes of super-diffusive and near sub-diffusive polariton transport associated with the fractal structure of the Monotile Hilbert space. We propose a strategy using resonantly excited polariton fluids to verify our findings.