The prediction of pressure drop for a droplet flow in a confined micro channel is presented using FE-FTM (Finite Element - Front Tracking Method). A single droplet is passing through 5:1:5 contraction - straight narrow channel - expansion flow domain. The pressure drop is investigated especially when the droplet flows in the straight narrow channel. We explore the effects of droplet size, capillary number (Ca), viscosity ratio ($chi$) between droplet and medium, and fluid elasticity represented by the Oldroyd-B constitutive model on the excess pressure drop (${Delta}p^+$) against single phase flow. The tightly fitted droplets in the narrow channel are mainly considered in the range of $0.001{leq}Ca{leq}1$ and $0.01{leq}{chi}{leq}100$. In Newtonian droplet/Newtonian medium, two characteristic features are observed. First, an approximate relation ${Delta}p^+{sim}{chi}$ observed for ${chi}{geq}1$. The excess pressure drop necessary for droplet flow is roughly proportional to $chi$. Second, ${Delta}p^+$ seems inversely proportional to Ca, which is represented as ${Delta}p^+{sim}Ca^m$ with negative m irrespective of $chi$. In addition, we observe that the film thickness (${delta}_f$) between droplet interface and channel wall decreases with decreasing Ca, showing ${delta}_f{sim}Ca^n$ Can with positive n independent of $chi$. Consequently, the excess pressure drop (${Delta}p^+$) is strongly dependent on the film thickness (${delta}_f$). The droplets larger than the channel width show enhancement of ${Delta}p^+$, whereas the smaller droplets show no significant change in ${Delta}p^+$. Also, the droplet deformation in the narrow channel is affected by the flow history of the contraction flow at the entrance region, but rather surprisingly ${Delta}p^+$ is not affected by this flow history. Instead, ${Delta}p^+$ is more dependent on ${delta}_f$ irrespective of the droplet shape. As for the effect of fluid elasticity, an increase in ${delta}_f$ induced by the normal stress difference in viscoelastic medium results in a drastic reduction of ${Delta}p^+$.