Living without internet really isn't too bad, particularly if you're 20 minutes away from a stable connection (library/McDonalds/Starbucks/etc).
I lived without internet a few years ago for about a year straight. I got really used to offline games, board games, doing things outside, getting better at playing guitar, and being with great people. If I ever needed internet, it was only a 20 minute drive away.
It's seriously not that bad. Sucks at first, but you quickly get used to it. Your girlfriend seems a little ignorant with how ingrained it is in your life, but I bet she's still awesome nonetheless. I dated a girl the same way - never had internet (and probably still doesn't), and she was pretty cool.
Internet is not like "running water." It's a luxury. Unless you need it for your job (which it sounds like you don't), you can get by without it.