We present a portable Rydberg-atom-based radio frequency (RF) electric (<inline-formula> <tex-math notation="LaTeX">E </tex-math></inline-formula>) field measurement instrument (Rydberg field measurement system or RFMS). The RFMS comprises an atomic RF field probe (RFP) connected by a ruggedized fiber-optic patch cord to a portable mainframe control unit. The unit includes a software interface providing self-calibrated SI-traceable RF measurement and analysis, including real-time field and measurement uncertainty readout and spectral RF waveform visualization. We characterize the RFP and measure polar field and polarization patterns along primary axes of the RFP at 12.6 GHz, obtained by rotations of the RFP in the far-field of a standard gain horn antenna. Field pattern measurements at 2.5 GHz are also presented. The measured field patterns are in good agreement with finite-element simulations of the RFP, establishing that the atom-based RF <inline-formula> <tex-math notation="LaTeX">E </tex-math></inline-formula>-field probe is well-suited for SI-traceable RF measurements over multiple bands. A one-time calibration procedure and an uncertainty analysis are introduced and implemented, as required for practical atomic probe designs to realize absolute-standard SI-traceable measurement capability. This includes uncertainty contributions from the atomic-spectroscopy measurement method and analysis and from the material, geometry, and hardware design. Simultaneous RF field and polarization measurements are demonstrated. RF waveform measurement is demonstrated using example pulsed and modulated RF signals.