Among natural disasters, earthquakes are recorded to have the highest rates of human loss in the past 20 years. Their unexpected nature has severe consequences on both human lives and material infrastructure, demanding urgent action to be taken. For effective emergency relief, it is necessary to gain awareness about the level of damage in the affected areas. The use of remotely sensed imagery is popular in damage assessment applications; however, it requires a considerable amount of labeled data, which are not always easy to obtain. Taking into consideration the recent developments in the fields of Machine Learning and Computer Vision, this study investigates and employs several Few-Shot Learning (FSL) strategies in order to address data insufficiency and imbalance in post-earthquake urban damage classification. While small datasets have been tested against binary classification problems, which usually divide the urban structures into collapsed and non-collapsed, the potential of limited training data in multi-class classification has not been fully explored. To tackle this gap, four models were created, following different data balancing methods, namely cost-sensitive learning, oversampling, undersampling and Prototypical Networks. After a quantitative comparison among them, the best performing model was found to be the one based on Prototypical Networks, and it was used for the creation of damage assessment maps. The contribution of this work is twofold: we show that oversampling is the most suitable data balancing method for training Deep Convolutional Neural Networks (CNN) when compared to cost-sensitive learning and undersampling, and we demonstrate the appropriateness of Prototypical Networks in the damage classification context.