There are really only one way for Arduino to communicate with your PC / server. That is by sending and recieving data over the serial line. To get the hit counter to Arduino you can either have some program on the PC / server send the data to Arduino, or you can have a program on the PC / server listening for a request from Arduino. But Arduino knwos nothing about such things as files or protocols or whatever. So you have to come up with a sort of simple protocol, a set of rules that the PC / server and Arduino agrees on.
If you can't program the PC your self, then the best solution will probably be to use something like MTTY to send the data to Arduino, and have the sketch on the Arduino do regular checks for the arrival of data over the serial line. Maybe first use soem kind of utility program to parse the file and extract just the numbr you need.
If you can program the PC yourself, then you can write a small program that will read the file with the hitcounter, parse it into a simple format, like the one suggested in Daniels post, and send it to Arduino, or let it wait for Arduino to ask for it.
A more advanced program, could monitor the hitcounter file and wait for changes and then kick in and do it's job.
If your PC is running Windows you could use a small VB script to regularly download the file from your server to your PC first. I bet that there are also LINUX utilities that can do that, but i don't know anything about that. Or maybe a scheduled FTP program could do that. A VB script would probably also be able to read the contents of the file and write out to a new file only the number you need to send to the Arduino.
I hope this sheds a little light on things