The brain-inspired Spiking Neural Networks (SNNs) have garnered considerable research interest due to their superior performance and energy efficiency in processing temporal signals. Recently, a novel multi-compartment spiking neuron model, namely the Two-Compartment LIF (TC-LIF) model, has been proposed and exhibited a remarkable capacity for sequential modelling. However, training the TC-LIF model presents challenges stemming from the large memory consumption and the issue of gradient vanishing associated with the Backpropagation Through Time (BPTT) algorithm. To address these challenges, online learning methodologies emerge as a promising solution. Yet, to date, the application of online learning methods in SNNs has been predominantly confined to simplified Leaky Integrate-and-Fire (LIF) neuron models. In this paper, we present a novel online learning method specifically tailored for networks of TC-LIF neurons. Additionally, we propose a refined TC-LIF neuron model called Adaptive TC-LIF, which is carefully designed to enhance temporal information integration in online learning scenarios. Extensive experiments, conducted on various sequential benchmarks, demonstrate that our approach successfully preserves the superior sequential modeling capabilities of the TC-LIF neuron while incorporating the training efficiency and hardware friendliness of online learning. As a result, it offers a multitude of opportunities to leverage neuromorphic solutions for processing temporal signals.