You do know the internet is more than just that box on your desk? If the phones don't work and the power in general is out, no way in hell is the internet going to be working.
No, I had no idea.
I doubt if the police station uses dial up. They either have cable or a satellite dish, which are independent of the phones.
But the cable and satellite companies would still need to be up and running. Even if some of the employees survived the Zombie Apocalypse, I doubt they would stay at work just to make sure the internet is running.