This paper deals with channel estimation and equalization, as well as noise variance estimation in uplink narrowband-internet of things (NB-IoT) system. Different techniques are studied in the context of NB-IoT, such as least square (LS) and linear minimum mean square error (LMMSE) for channel estimation, and zero forcing (ZF) and MMSE for equalization. It is shown that a low-complexity application of MMSE-based methods is made possible in NB-IoT by taking advantage of the small number of subcarriers. Furthermore , a noise variance estimator is suggested based on the combination of two successive observations of pilots, assuming slowly varying channel. We also prove that the proposed estimator is efficient, and confirm by simulations that both LMMSE channel estimator and MMSE equalizer can use the estimated noise variance instead of the exact value without loss of performance.