I have to ask first, did you put the batteries in correctly? Check that the polarity is correct. If they're in the wrong way the meter will not move.
I have a Nikon EM which I purchased a few weeks ago. I had the same problem as you describe -- the needle would not budge. I finally figured out that if the light level that the camera is sensing is out of its range the needle will not move. The EM stays at the top above 1/1000th regardless. Take the camera outside on a sunny day and setting a few different film speeds. Move the aperture ring through its settings for each film speed. You should be able to get the meter to move if its working.
If you like the camera enough to get it repaired then I would recommend Garry's Camera. They fixed a Minolta for me and did a nice job even recalibrating for a silver oxide battery instead of the mercury battery. They charge a flat fee and for a Nikon FG it would be $45 + $8 shipping. Their site is www22.brinkster.com/garryscamera.