The Rare-RI Ring (R3) is a recently commissioned cyclotronlike storage ring mass spectrometer dedicated to mass measurements of exotic nuclei far from stability at Radioactive Isotope Beam Factory (RIBF) in RIKEN. The first application of mass measurement using the R3 mass spectrometer at RIBF is reported. Rare isotopes produced at RIBF-^{127}Sn, ^{126}In, ^{125}Cd, ^{124}Ag, ^{123}Pd-were injected in R3. Masses of ^{126}In, ^{125}Cd, and ^{123}Pd were measured whereby the mass uncertainty of ^{123}Pd was improved. This is the first reported measurement with a new storage ring mass spectrometry technique realized at a heavy-ion cyclotron and employing individual injection of the preidentified rare nuclei. The latter is essential for the future mass measurements of the rarest isotopes produced at RIBF. The impact of the new ^{123}Pd result on the solar r-process abundances in a neutron star merger event is investigated by performing reaction network calculations of 20 trajectories with varying electron fraction Y_{e}. It is found that the neutron capture cross section on ^{123}Pd increases by a factor of 2.2 and β-delayed neutron emission probability, P_{1 n}, of ^{123}Rh increases by 14%. The neutron capture cross section on ^{122}Pd decreases by a factor of 2.6 leading to pileup of material at A=122, thus reproducing the trend of the solar r-process abundances. The trend of the two-neutron separation energies (S_{2n}) was investigated for the Pd isotopic chain. The new mass measurement with improved uncertainty excludes large changes of the S_{2n} value at N=77. Such large increase of the S_{2n} values before N=82 was proposed as an alternative to the quenching of the N=82 shell gap to reproduce r-process abundances in the mass region of A=112-124.