In addition to this ^^^ ...
You didn't mention aperture setting. To get a correct exposure, the shutter speeds works together with the aperture to determine how much light gets to the film. If your camera on Auto set the shutter speed so fast, chances are your aperture was wide open, especially using 160ISO film. Did you set the camera to the correct ISO?
The focal length of the lens doesn't matter except when considering what is included in the frame. For example, it might be a sunny day but if you are aiming your 135mm at a scene totally in the shadows, then it will give a different exposure reading. If you were shooting with a wider lens, you might end up with more lighted areas in the frame and thus requiring a different exposure.
But really, that's already getting beyond the discussion.
First question: If 1/2000th was not the correct setting, it's more likely to be underexposed than overexposed. That is a fast shutter speed, which means it's letting in less light than 1/500th. Your exposures at 1/500th may be overexposed if you did not change the aperture setting. But as JC said, we have no way of knowing since we don't know what you were shooting.
Second: If you were shooting 160ISO, it was likely Kodak Portra. I don't know of any Fuji films that come rated as 160 - just 100, 200, or 400. Someone else can correct me if I'm wrong about that.
Third: I agree - get a light meter or use a light meter app on your smartphone if you have one. Or you can brush up on the Sunny 16 guidelines for exposure (google it.)
Finally, when you get your shots developed, post some for more directed, specific feedback about exposure. And if you really want to be systematic about it, don't forget to write down the settings for each shot that you take so you know what settings produced each image.