  We use transport and neutron scattering to study the electronic phase diagram
and spin excitations of NaFe$_{1-x}$Cu$_x$As single crystals. Similar to Co-
and Ni-doped NaFeAs, a bulk superconducting phase appears near $x\approx2\%$
with the suppression of stripe-type magnetic order in NaFeAs. Upon further
increasing Cu concentration the system becomes insulating, culminating in an
antiferromagnetically ordered insulating phase near $x\approx 50\%$. Using
transport measurements, we demonstrate that the resistivity in
NaFe$_{1-x}$Cu$_x$As exhibits non-Fermi-liquid behavior near $x\approx1.8\%$.
Our inelastic neutron scattering experiments reveal a single neutron spin
resonance mode exhibiting weak dispersion along $c$-axis in
NaFe$_{0.98}$Cu$_{0.02}$As. The resonance is high in energy relative to the
superconducting transition temperature $T_{\rm c}$ but weak in intensity,
likely resulting from impurity effects. These results are similar to other iron
pnictides superconductors despite the superconducting phase in
NaFe$_{1-x}$Cu$_x$As is continuously connected to an antiferromagnetically
ordered insulating phase near $x\approx 50\%$ with significant electronic
correlations. Therefore, electron correlations is an important ingredient of
superconductivity in NaFe$_{1-x}$Cu$_x$As and other iron pnictides.
