Sure you could, but that would mean you had to boot the computer from your USB stick so it ran the Linux instead of whatever (Windows?) they offer you there. That faces two big problems:
1) Very probably (at least if they've given this a thought) you cannot boot your own operating system off a cd/external hd/usb stick/other media; if you could, it meant you could do all sorts of naughty things. So probably booting off anything else than the computer's own harddisk is disabled, and you can't change that.
2) Even if you could boot, you probably couldn't get anywhere further than that machine; the network is most probably secured, and requires some sort of a login (to a domain if I'm right) to access the internet, for example. The operating system that runs there normally is configured to do this automatically, but your fancy Linux probably doesn't. And I guess they don't tell you how to connect your own operating system to their network
There is a reason why they don't want you to download things there. It's a public pc, so you just have to live by their rules; if you need to download something so badly, get a laptop (you can get Asus Eee Pc laptops for less than 150€ today) and either connect it wirelessly (or by wire if you can) to the internet or trough your mobile phone or something. Or just wait until you get home.