The integration of robotics and augmented reality (AR) holds transformative potential for advancing human-robot interaction (HRI), offering enhancements in usability, intuitiveness, accessibility, and collaborative task performance. This paper introduces and evaluates a novel multimodal AR-based robot puppeteer framework that enables intuitive teleoperation via virtual counterpart through large language model (LLM)-driven voice commands and hand gesture interactions. Utilizing the Meta Quest 3, users interact with a virtual counterpart robot in real-time, effectively "puppeteering" its physical counterpart within an AR environment. We conducted a within-subject user study with 42 participants performing robotic cube pick-and-place with pattern matching tasks under two conditions: gesture-only interaction and combined voice-and-gesture interaction. Both objective performance metrics and subjective user experience (UX) measures were assessed, including an extended comparative analysis between roboticists and non-roboticists. The results provide key insights into how multimodal input influences contextual task efficiency, usability, and user satisfaction in AR-based HRI. Our findings offer practical design implications for designing effective AR-enhanced HRI systems.