How are you trying to update your software? The easiest way is to select 'Software Update...' from the Apple menu in the top left corner of your screen. This does require an internet connection - I presume you were able to connect to the internet on your Mac Mini in order to post this. Does Software Update give any error messages?
Also, could your software be up to date already? What versions are you running? To get the version of Mac OS X that you're running, select 'About This Mac' from the Apple menu, and the version should be displayed immediately below the Apple logo and "Mac OS X". The latest version at time of writing is 10.6.6 - if you have that, you won't get any updates to Mac OS X, because there aren't any! Similarly, iTunes is at version 10.1.1, and will be updated by Software Update if you have an earlier version.