The puzzling electronic correlation effects in the sodium cobaltate system are studied by means of the combination of density-functional theory and the rotationally invariant slave boson (RISB) method in a cellular-cluster approach. Realistic nonlocal correlations are hence described in the short-range regime for finite Coulomb interactions on the underlying frustrated triangular lattice. A local Hubbard U is sufficient to model the gross in-plane magnetic behavior with doping x, namely, antiferromagnetic correlations at intermediate doping and the onset of ferromagnetic order above x>3/4 with a mixed phase for 0.62<x<3/4. Important insight is thereby provided by the occupations of local cluster multiplets retrieved from the RISB framework. The extended modeling of the x >= 2/3 doping regime with an additional intersite Coulomb repulsion V on an experimentally verified effective kagome lattice allows to account for relevant charge-ordering physics. Therewith a fluctuating charge-density-wave state with small quasiparticle weight and a maximum in-plane magnetic susceptibility may be identified at x similar to 3/4, just where the magnetic ordering sets in.