This is a Windows 2003 (or later) machine that gets switched on every morning, but no one logs on until some hours later.
I want to use the time in between to run a back
Short answer: GPEDIT.MSC (Start, Run, GPEdit.msc)
Windows 2000 and above [1] offer a computer Startup Scripts collection in the policy editor:
There's an equivalent logon script area (i.e. after computer startup, when a user logs on) in the User configuration bit.
Longer:
GPEDIT.MSC is the Group Policy editing console, and runs against the local computer's Local Group Policy store when it's used directly, so it's useful for setting local-only parameters. When using Active Directory, a similar interface is used to edit AD Forest-hosted group policy objects (GPOs), so the same settings are available across a bunch of machines.
The computer startup scripts run in the computer context, i.e. as LocalSystem, as you noted, so they often can't access network drives which require a certain user or group membership to work. When domain-member computers access network resources, they generally (with exceptions) authenticate using their MACHINENAME$ account - which lets you set Share/NTFS permissions to allow a computer to access those resources.
A startup script is a quick and easy way of getting a process running when the machine boots.
The computer startup process will be affected by the time it takes to run the program, though, so you might want to ensure you call it with the START command from a batch file, or specifying not to wait for the executable to complete in whatever script language you use. (the key point there is: run the script asynchronously unless it's critical, or doesn't need to be run asynchronously cos it will always take no time at all. Long boots = unhappy users).
Using a Win32 Service is an alternative option - you can use the SRVANY utility from the Resource Kit to "service-ify" pretty much any executable. VS.Net 2002 and later also let you build a managed service directly.
And Task Scheduler gets much more capable as of Vista/2008, able to run scripts at startup, on idle, and/or when Event Logs are generated or certain other conditions are met: it's pretty cool! Scheduled Tasks has the possible advantage of being able to specify the user account under which the task runs, if that's important to you.
Caveat Scriptor: http://support.microsoft.com/kb/256320
Run Startup Scripts Asynchronously: http://msdn.microsoft.com/en-us/library/ms811602.aspx
Vista Task Scheduler (what's new): http://technet.microsoft.com/en-us/appcompat/aa906020.aspx
[1] Windows XP, 2003, Vista/2008, Windows 7/2008R2, Windows 8/2012, Windows 8.1/2012R2, Windows 10/Windows Server 2016. Everything. But NT4 didn't!