Narrowband Internet of Things (NB-IoT) over non-terrestrial networks (NTN) is a key enabler for massive Internet of Things (IoT) in 6G, but in low Earth orbit (LEO) scenarios, large and time-varying Doppler shifts generate carrier frequency offset (CFO) beyond the correction range of standard user equipment (UE), making initial downlink synchronization a major bottleneck. This paper analyzes Doppler characteristics in realistic NB-IoT LEO scenarios, reviews Doppler mitigation strategies, and proposes a standard-compliant, low-overhead search-space optimization method for downlink acquisition. Results under realistic LEO conditions with real-time measurements show reduced acquisition overhead while maintaining synchronization reliability, supporting NB-IoT adaptation to 6G NTN deployment.