Why can't I update my device? I have a Galaxy S3 (GT-I9300), my current Android version is 4.0.4.

I am trying to define software updates for it, via the Software Updates screen. (Settings > About).

However, I get "Unable to connect to network" every time. What network is it trying to connect? Can I define it? If so, what should I define it too?

Also, will software updates allow me to update to 4.1.2? If not, how can I?

Can anyone please point to a resource which explains it? (Or even better, answer it here).

If I'm lacking any technical details, please leave a comment and I'll edit my question right away.
May be I am too easy with you. 
But just for the fact of helping, were you able to access internet through your mobile? 
If the answer is 'Yes' then there must be a bug with your OS. (There are possibilities for that in one or two devices)
My suggestion is to use 'Kies' from a PC.
Last week only I got to know even GT-I9300 devices are network locked in some countries. A friend of mine's GT-I9100 started asking for Network Unlock code all of a sudden. After a master reset, we noticed some new apps from Vodafone and the default language turned into Spanish or something. 
So I think your GS3 might be trying to access its original operator. 
Or may be some other reason. 
Just try my suggestions if you didn't do already.