Is it possible to have my minecraft server update itself?

You can use McSvSnapshotUpdater to accomplish this, you can download it here.

Once you download it, you can extract the files in the zip file to the server directory, then every time you launch it, it'll update the file "minecraftsnapshotserver.jar" to the latest version.

If you launch the server through a bat file, you can edit it to look like this, and everytime you launch it, it'll check for updates and download the newest version (if there is any) and will launch that:

title run-minecraft
mcsvsu.exe
java -Xms1024M -Xmx2048M -jar minecraftsnapshotserver.jar nogui
pause

Or if you want to rename the server before running it, (replace newnameoffile with what you want)

title run-minecraft
mcsvsu.exe
if exist newnameoffile.jar del newnameoffile.jar
ren minecraftsnapshotserver.jar newnameoffile.jar
java -Xms1024M -Xmx2048M -jar newnameoffile.jar nogui
pause

Disclaimer: This software is developed, maintained and published by me.

Sadly, not without some programming. You have to manually go to minecraft.net and download the latest server.