Non-orthogonal multiple access (NOMA) and ambient backscatter communication (AmBC) play major roles to enhance spectrum efficiency in wireless communication systems. Besides, the AmBC provides good reinforcement for the current trend towards dispensing batteries for battery-free Internet-of-Things (IoT) devices. In this paper, we propose a symbiotic battery-free IoT system, that exploits the downlink transmission of a NOMA multiplexing enabled cellular network, to permit an IoT spectrum-efficient uplink communication. The IoT backscatter device (BD) performs a symbiotic radio (SR) relation with the cellular source to power its communication by intelligently reflecting the received power. We derive a closed-form expression of the ergodic capacity (EC) of the BD transmission and tight approximations of the ECs of the cellular source transmission, where all channels undergo Nakagami-m fading. Additionally, we validate the analytical results obtained using Monte-Carlo simulations. The influences of several system parameters such as power allocation factor, reflection coefficient, and channels’ severity factors have been investigated. Finally, the performance of the proposed system is compared with a benchmark OMA-based system to highlight the achievable performance improvement.