We have used muon spin rotation and relaxation (μSR) and ²³Na nuclear magnetic resonance (NMR) spectroscopic methods in the NaOsO₃ antiferromagnetic phase to determine the temperature evolution of the magnetic order parameter and the role of the magnetic fluctuations at the Néel temperature. Additionally, we performed muon spin relaxation measurements in the vicinity of T_A = 30 K, where the appearance of an anomaly in the electrical resistivity was suggested to be due to a progressive reduction of the Os magnetic moment associated with spin fluctuation. Our measurements suggest the absence of prominent change in the spin fluctuations frequency at T_A, within the muon probing time scale and the absence of a reduction of the localized Os magnetic moment reflected by the stability within few permille of the local magnetic field strength sensed by the muons below 50 K.