  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 $x\simeq0.375$ 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.
