  We report results of magnetization and $^{31}$P NMR measurements under high
pressure up to 6.4~GPa on RbMoOPO$_4$Cl, which is a frustrated square-lattice
antiferromagnet with competing nearest-neighbor and next-nearest-neighbor
interactions. Anomalies in the pressure dependences of the NMR shift and the
transferred hyperfine coupling constants indicate a structural phase transition
at 2.6~GPa, which is likely to break mirror symmetry and triggers significant
change of the exchange interactions. In fact, the NMR spectra in magnetically
ordered states reveal a change from the columnar antiferromagnetic (CAF) order
below 3.3~GPa to the N\'{e}el antiferromagnetic (NAF) order above 3.9~GPa. The
spin lattice relaxation rate $1/T_1$ also indicates a change of dominant
magnetic fluctuations from CAF-type to NAF-type with pressure. Although the NMR
spectra in the intermediate pressure region between 3.3 and 3.9 GPa show
coexistence of the CAF and NAF phases, a certain component of $1/T_1$ shows
paramagnetic behavior with persistent spin fluctuations, leaving possibility
for a quantum disordered phase. The easy-plane anisotropy of spin fluctuations
with unusual nonmonotonic temperature dependence at ambient pressure gets
reversed to the Ising anisotropy at high pressures. This unexpected anisotropic
behavior for a spin 1/2 system may be ascribed to the strong spin-orbit
coupling of Mo-4$d$ electrons.
