Supercapacitors (SCs) are widely recognized as excellent clean energy storage devices. Accurate state of health (SOH) estimation and remaining useful life (RUL) prediction are essential for ensuring their safe and reliable operation. This paper introduces a novel method for SOH estimation and RUL prediction, based on a hybrid neural network optimized by an improved honey badger algorithm (HBA). The method combines the advantages of convolutional neural network (CNN) and bidirectional long-short-term memory (BiLSTM) neural network. The HBA optimizes the hyperparameters of the hybrid neural network. The CNN automatically extracts deep features from time series data and reduces dimensionality, which are then used as input for the BiLSTM. Additionally, recurrent dropout is introduced in the recurrent layer to reduce overfitting and facilitate the learning process. This approach not only improves the accuracy of estimates and forecasts but also significantly reduces data processing time. SCs under different working conditions are used to validate the proposed method. The results show that the proposed hybrid model effectively extracts features, enriches local details, and enhances global perception capabilities. The proposed hybrid model outperforms single models, reducing the root mean square error to below 1%, and offers higher prediction accuracy and robustness compared to other methods.