At the microscale, concrete is a heterogeneous particulate composite comprising multiple solid phases. One of the most critical of these is the Interfacial Transition Zone (ITZ), which exhibits reduced stiffness due to its relatively high porosity. Under mechanical loading, cracks often originate in the ITZ and then propagate into the surrounding phases. Consequently, concrete fracture is characterized by complex, highly nonlinear dynamics, making it computationally demanding to resolve these processes using conventional numerical methods. Deep neural networks (DNNs) have emerged as a promising way to reduce the cost of computationally intensive simulations, including fracture analysis in concrete microstructures. In this study, we develop and compare two DNN-based approaches for predicting the spatiotemporal evolution of fractures in concrete microstructures. The first approach relies on convolutional data-driven surrogate modelling, while the second leverages Neural Operator (NO) frameworks. First, we benchmark NO models against established data-driven architectures, such as U-Net, to evaluate their ability to reproduce the full fracture dynamics. Second, the focus is on checking if the main benefits often given to NOs, like learning mappings over function spaces to decrease data dependence and allowing zero-shot generalization across resolutions, are still effective for the strongly time-dependent and non-linear fracture processes seen in concrete. The training and evaluation data are generated in Abaqus/CAE and treated as the ground truth. The dataset specifically consists of finite element phase-field fracture simulations of concrete microstructures, which provide time-resolved fields describing crack initiation and propagation.