You cannot use Disc Sharing to watch a DVD movie.
This document outlines the provision.
http://support.apple.com/kb/TS2057
To quote from the document:
MacBook Air and Mac mini do not support playing DVD movies or audio CD media with Remote Disc. In addition, some discs, such as game discs, may be copy-protected and are therefore unusable. To play DVD movies, audio CD media, or copy-protected discs on the MacBook Air or Mac mini, use the MacBook Air SuperDrive.
Your choices are this.
1) Use MBA Superdrive
2) Use another compatible external DVD drive on the MBA
3) Use the Mac Mini to rip the DVD using software designed for the task. And then use VLC or iTunes on the MBA to watch said ripped file (depending on file type the DVD is converted to). Legal and copy-protection conversion restrictions apply.