Analog in-memory computing (AIMC) cores offers significant performance and energy benefits for neural network inference with respect to digital logic (e.g., CPUs). AIMCs accelerate matrix-vector multiplications, which dominate these applications' run-time. However, AIMC-centric platforms lack the flexibility of general-purpose systems, as they often have hard-coded data flows and can only support a limited set of processing functions. With the goal of bridging this gap in flexibility, we present a novel system architecture that tightly integrates analog in-memory computing accelerators into multi-core CPUs in general-purpose systems. We developed a powerful gem5-based full system-level simulation framework into the gem5-X simulator, ALPINE, which enables an in-depth characterization of the proposed architecture. ALPINE allows the simulation of the entire computer architecture stack from major hardware components to their interactions with the Linux OS. Within ALPINE, we have defined a custom ISA extension and a software library to facilitate the deployment of inference models. We showcase and analyze a variety of mappings of different neural network types, and demonstrate up to 20.5x/20.8x performance/energy gains with respect to a SIMD-enabled ARM CPU implementation for convolutional neural networks, multi-layer perceptrons, and recurrent neural networks.