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We analyze the time evolution of several physical observables, namely the pressure anisotropy, the scalar condensate, the charge density, and also, for the first time, the non-equilibrium entropy for a Bjorken expanding strongly coupled $\mathcal{N}=4$ Supersymmetric Yang-Mills plasma charged under an Abelian $U(1)$ subgroup of the global $SU(4)$ R-symmetry. This represents a far-from-equilibrium, hot and dense strongly coupled quantum fluid with a critical point in its phase diagram. For some sets of initial data preserving all the energy conditions, dynamically driven transient violations of the dominant and the weak energy conditions are observed when the plasma is still far from the hydrodynamic regime. The energy conditions violations get stronger at larger values of the chemical potential to temperature ratio, $μ/T$, indicating that those violations become more relevant as the strongly coupled quantum fluid approaches its critical regime. For some of those energy conditions violations, it is observed a clear correlation with different plateau structures formed in the far from equilibrium entropy, indicating the presence of transient, early time windows where the Bjorken expanding plasma has zero entropy production even while being far from equilibrium. The hydrodynamization of the pressure anisotropy and also the much later thermalization of the scalar condensate are generally found to be delayed, within small relative tolerances, as $μ/T$ is increased towards criticality. The value of $μ/T$ in the medium is enhanced by increasing its initial charge density, and/or also by reducing its initial energy density.