We present our work in updating the spin and orbital parameters for the newly discovered accreting millisecond X-ray pulsar (AMXP) IGR J17591-2342 through pulsar timing and analyzing its energy dependent pulse behaviors. The data being analyzed were collected by Neutron Star Interior Composition ExploreR, which observed this AMXP from 2018 August to October. Using the pulse arrival time delay technique, more accurate spin and orbital parameters were evaluated. From the measured spin frequency derivative, it is estimated that the magnetic field of the neutron star in IGR J17591-2342 is approximately 4 × 108 G. Precise pulse profiles can be made using the updated spin and orbital parameters. The soft phase lag phenomenon that is usually seen in other AMXPs is also observed from ∼4 to 12 keV with a value of 0.06 cycles (1.14 μs). Additionally, the pulsed fractional amplitude increases from 1 to ∼5 keV and then decreases for higher energy bands. We found that these phenomena, as well as the energy spectrum, can be explained by the two-component model with a relatively strong blackbody component and an additional unpulsed disk blackbody component.