Side-channel disassembly attacks recover CPU instructions from power or electromagnetic side-channel traces measured during code execution. These attacks typically rely on physical access, proximity to the victim device, and high sampling rate measuring instruments. In this work, however, we analyze the CPU instruction-level power side-channel leakage in an environment that lacks physical access or expensive measuring equipment. We show that instruction leakage is present even in a multitenant FPGA scenario, where the victim uses a soft-core CPU, and the adversary deploys on-chip voltage-fluctuation sensors. Unlike previous remote power side-channel attacks, which either require a considerable number of victim traces or attack large victim circuits such as machine learning accelerators, we take an evaluator’s point of view and provide an analysis of the instruction-level power side-channel leakage of a small open-source RISC-V soft processor core. To investigate whether the power side-channel traces leak secrets, we profile the victim device and implement various instruction opcode classifiers based on both classical machine learning algorithms used in disassembly attacks, and novel, deep learning approaches. We explore how parameters such as placement, trace averaging, profiling templates, and different FPGA families (including a cloud-scale FPGA) impact the classification accuracy. Despite the limited leakage of the soft-core CPU victim and a reduced accuracy and sampling rate of on-chip sensors, we show that in a worst-case scenario for the evaluator, i.e., an attacker breaching physical separation, we can identify the opcode of executed instructions with an average accuracy as high as 86.46%. Our analysis shows that determining the executed instruction type is not a classification bottleneck, while leakages between instructions of the same type can be challenging for deep learning models to distinguish. We also show that the instruction-level leakage is significantly reduced in a cloud-scale FPGA scenario with higher soft-core CPU frequencies. Nevertheless, our results show that even small circuits, such as soft-core CPUs, leak potentially exploitable information through on-chip power side channels, and users should deploy mitigation techniques against disassembly attacks to protect their proprietary code and data.