We present a study of the radio variability of bright, S_1.4_>=100mJy, high-redshift quasars at z?=3 on timescales of up to 30-40 yrs. The study involved simultaneous RATAN-600 measurements at the frequencies of 2.3, 4.7, 8.2, 11.2, and 22.3GHz in 2017-2020. In addition, data from the literature were used. We have found that the variability index, V_S_, which quantifies the normalized difference between the maximum and minimum flux density while accounting for measurement uncertainties, ranges from 0.02 to 0.96 for the quasars. Approximately half of the objects in the sample exhibit a variability index within the range from 0.25 to 0.50, which is comparable to that observed in blazars at lower redshifts. The distribution of V_S_ at 22.3GHz is significantly different from that at 2.3-11.2GHz, which may be attributed to the fact that a compact AGN core dominates at the source's rest frame frequencies greater than 45GHz, leading to higher variability indices obtained at 22.3GHz (the V_S_ distribution peaks around 0.4) compared to the lower frequencies (the V_S_ distribution at 2.3 and 4.7GHz peaks around 0.1-0.2). Several source groups with distinctive variability characteristics were found using the cluster analysis of quasars. We propose seven new candidates for gigahertz-peaked spectrum (GPS) sources and five new megahertz-peaked spectrum (MPS) sources based on their spectrum shape and variability features. Only 6 out of the 23 sources previously reported as GPS demonstrate a low variability level typical of classical GPS sources (V_S_<0.25) at 4.7-22.3GHz. When excluding the highly variable peaked-spectrum blazars, we expect no more than 20% of the sources in the sample to be GPS candidates and no more than 10% to be MPS candidates.