We study the effects of electron-electron interactions and hole doping on the electronic structure of Cu-doped NaFeAs using the density functional theory plus dynamical mean-field theory (DFT+DMFT) method. In particular, we employ an effective multi-orbital Hubbard model with a realistic bandstructure of NaFeAs in which Cu-doping was modeled within a rigid band approximation and compute the evolution of the spectral properties, orbital-dependent electronic mass renormalizations, and magnetic properties of NaFeAs upon doping with Cu. In addition, we perform fully charge self-consistent DFT+DMFT calculations for the long-range antiferromagnetically ordered Na(Fe,Cu)As with Cu $x=0.5$ with a real-space ordering of Fe and Cu ions. Our results reveal a crucial importance of strong electron-electron correlations and local potential difference between the Cu and Fe ions for understanding the textbf{k}-resolved spectra of Na(Fe,Cu)As. Upon Cu-doping, we observe a strong orbital-dependent localization of the Fe $3d$ states accompanied by a large renormalization of the Fe $xy$ and $xz$/$yz$ orbitals. Na(Fe,Cu)As exhibits bad metal behavior associated with a coherence-to-incoherence crossover of the Fe $3d$ electronic states and local moments formation near a Mott metal-insulator transition (MIT). For heavily doped NaFeAs with Cu $x sim 0.5$ we obtain a Mott insulator with a band gap of $sim$0.3 eV characterized by divergence of the quasiparticle effective mass of the Fe $xy$ states. In contrast to this, the quasiparticle weights of the Fe $xz$/$yz$ and $e_g$ states remain finite at the MIT. The MIT occurs via an orbital-selective Mott phase to appear at Cu $xsimeq0.375$ with the Fe $xy$ states being Mott localized. We propose the possible importance of Fe/Cu disorder to explain the magnetic properties of Cu-doped NaFeAs.
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