exec master..xp_cmdshell 'copy C:\file.txt \\servershare\file.txt'

xp_cmdshell allows you to run dos command prompts. If you are using 2005 you will need to check that this is enabled as it is...

I would use xp_cmdshell

Alternatively if you feel more comfortable using the GUI you can create a maintenance plan to backup databases which can have a step in it to determine how many days...

It depends on support arrangements in place between your company and theirs. If they insist on more than read / write access to the database and need sysadmin rights...