The electronic structure and dielectric screening of finite-length armchair carbon nanotubes are studied with both tight-binding and ab initio methods. Good agreement is found in the band gap oscillation patterns and dielectric constants, which validates the tight-binding method as a reliable and fast approach to describe the screening effect of carbon nanotubes. For an illustration, our method is applied to a system consisting of a short (6,6) nanotube filled with six water molecules. Substantial screening of the water dipoles through the nanotube is observed. This polarization effect should have an important influence on the permeation of water and other biomolecules inside carbon nanotubes.