Embedded real-time systems must respond to external events within tightly bounded timeframes to ensure safety, correctness, and reliability. While Real-Time Operating Systems (RTOSes) ease the development of complex applications by providing abstractions for multi-tasked execution, they introduce overheads in the form of task switching latency and jitter, which impact timing predictability. Minimizing these effects is essential for reducing response times and enabling robust worst-case timing analysis. We present RTOSUnit, a configurable hardware acceleration unit designed to reduce context-switch latency and jitter in embedded real-time systems. By integrating the RTOSUnit into three RISC-V cores of varying complexity, we demonstrate its portability. Through a range of configurations, from lightweight scheduling acceleration to full context-switch and -preloading support, RTOSUnit achieves up to 76% reduction in mean context-switch latency and can be configured to completely eliminate jitter on selected cores. Area overheads in 22nm ASIC implementations range from negligible (within EDA tool heuristics noise) to 44\,%, with all configurations maintaining viable operating frequencies and power envelopes suitable for embedded systems. RTOSUnit offers a flexible and efficient, open-source. https://github.com/esa-tu-darmstadt/RTOSUnit_Integration foundation for hardware-assisted real-time scheduling, paving the way for broader integration into future embedded SoCs.