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 multiorbital Hubbard model with a realistic band structure of NaFeAs in which Cu-doping was modeled within a rigid band approximation and compute the evolution of the spectral properties, orbital-selective electronic mass renormalizations, and magnetic properties of NaFeAs on 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 k-resolved spectra of Na(Fe,Cu)As. On Cu-doping, we observe a strong orbital-selective 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∼0.5 we obtain a Mott insulator with a band gap of ∼0.3 eV which is 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 states remain finite at the MIT. The MIT occurs via an orbital-selective Mott phase to appear at Cu x≃0.375 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.