Aboveground biomass (AGB) is an important indicator for crop-growth monitoring and yield prediction, and accurate monitoring of AGB is beneficial to agricultural fertilization management and optimization of planting patterns. Imaging spectrometer sensors mounted on unmanned aerial vehicle (UAV) remote-sensing platforms have become an important technical method for monitoring AGB because the method is convenient, rapidly collects data and provides image data with high spatial and spectral resolution. To confirm the feasibility of UAV hyperspectral remote-sensing technology to estimate AGB, this study acquired hyperspectral images and measured AGB data over the potato bud, tuber formation, tuber growth, and starch-storage periods. The canopy spectrum obtained in each growth period was smoothed by using the Savitzky–Golay filtering method, and the spectral-reflection feature parameters, spectral-location feature parameters, and vegetation indexes were extracted. First, a Pearson correlation analysis was performed between the three types of characteristic spectral parameters and AGB, and the spectral parameters that reached a significant level of 0.01 in each growth period were selected. Next, the spectral parameters reaching a significance of 0.01 were optimized and screened by moving window partial least squares (MWPLS), Monte Carlo uninformative variable elimination (MC-UVE), and random frog (RF) methods, and the final model parameters were determined according to the thresholds of the root mean square error of cross-validation (RMSEcv), the reliability index, and the selected probability. Finally, the three optimal characteristic spectral parameters and their combinations were used to estimate the potato AGB in each growth period by combining the partial least squares regression (PLSR) and Gaussian process regression (GPR) methods. The results show that, (i) ranked from high to low, vegetation indexes, spectral-location feature parameters, and spectral-reflection feature parameters in each growth period are correlated with the AGB, and these correlations all first improve and then degrade in going from the budding period to the starch-storage period. (ii) The AGB estimation model based on the characteristic variables screened by the three methods in each growth period is most accurate with RF, less so with MC-UVE, and least accurate with MWPLS. (iii) Estimating the AGB with the same variables combined with the PLSR method in each growth period is more accurate than the corresponding GPR method, but the estimations produced by the two methods both show a trend of first improving and then worsening from the budding period to the starch-accumulation period. The accuracy of the estimation models constructed by PLSR and GPR from high to low is based on comprehensive variables, vegetation indexes, spectral-location feature parameters and spectral-reflection feature parameters. (iv) When combined with the RF-PLSR method to estimate AGB in each growth period, the best R2 values are 0.65, 0.68, 0.72, and 0.67, the corresponding RMSE values are 167.76, 162.98, 160.77, and 169.24 kg/hm2, and the corresponding NRMSE values are 19.76%, 16.01%, 15.04%, and 16.84%. The results of this study show that a variety of characteristic spectral parameters may be extracted from UAV hyperspectral images, that the RF method may be used for optimizing and screening, and that PLSR regression provides accurate estimates of the potato AGB. The proposed approach thus provides a rapid, accurate, and nondestructive way to monitor the growth status of potatoes.