Energy and data-efficient online time series prediction for predicting evolving dynamical systems are critical in several fields, especially edge AI applications that need to update continuously based on streaming data. However, current Deep Neural Network (DNN)-based supervised online learning models require a large amount of training data and cannot quickly adapt when the underlying system changes. Moreover, these models require continuous retraining with incoming data making them highly inefficient. We present a novel Continuous Learning-based Unsupervised Recurrent Spiking Neural Network Model (CLURSNN), trained with spike timing dependent plasticity (STDP) to solve these issues. CLURSNN makes online predictions by reconstructing the underlying dynamical system using Random Delay Embedding by measuring the membrane potential of neurons in the recurrent layer of the recurrent spiking neural network (RSNN) with the highest betweenness centrality. We also use topological data analysis to propose a novel methodology using the Wasserstein Distance between the persistent homologies of the predicted and observed time series as a loss function. We show that the proposed online time series prediction methodology outperforms state-of-the-art DNN models when predicting an evolving Lorenz63 dynamical system.