I had an indoor shoot in low light so I broke out my Nikon FE and SB-15 flash. I used Portra 400 for half the shoot and Portra 160 for the other half. I had everything set correctly on my camera and flash except that I forgot to set my camera to 1/125 and slower, as that's the FE's sync speed. I had the shutter speed set to Auto. I had the flash set to Auto. The SB-15 wasn't blinking red after I shot so I know that the pictures weren't underexposed. My concern is whether the pictures will come out overexposed or is it ok to use a flash when in Auto mode on your camera?
The style for todays cameras is that they know to Not allow shutter speed faster than maximum sync speed if a flash is detected present. Since your FE has Auto, it should also know to limit shutter speed for flash. Ken Rockwell says it does:
The FE switches to 1/125 when a flash is ON and the FE is set to AUTO. The meter needle shows the ambient light, making it easy to set light ratios.
I think you're OK with it.