We report the discovery from our IRSF/SIRIUS Near-Infrared (NIR) variability survey of five new classical Cepheids located in the inner Galactic Disk, at longitude $lsimeq -40^{circ}$. The new Cepheids are unique in probing the kinematics and metallicity of young stars at the transition between the inner Disk and the minor axis of the central Bar, where they are expected to be less affected by its dynamical influence.This is also the first time that metallicity of Cepheids is estimated on the basis of medium-resolution ($Rsim3,000$) NIR spectra, and we validated our results with data in the literature, finding a minimal dependence on the adopted spectroscopic diagnostics. This result is very promising for using Cepheids as stellar proxy of the present-time chemical content of the obscured regions in the Disk. We found that the three Cepheids within 8--10 kpc from us have metallicities consistent with the mean radial metallicity gradient, and kinematics consistent with the Galactic rotation curve. Instead, the closest ($sim$4 kpc)/farthest ($sim$12 kpc) Cepheids have significant negative/positive residuals, both in velocity and in iron content. We discuss the possibility that such residuals are related to large-scale dynamical instabilities, induced by the bar/spiral-arm pattern, but the current sample is too limited to reach firm conclusions.