How Important is the 180 Degree Shutter Angle Rule? I do notice that the 180 degree shutter angle look better, and I try to achieve it whenever I could. However, when shooting in bright conditions, I could not achieve it without a variable ND filter, which is too expensive for me. Even the cheap ones are expensive, but they are pretty bad.

I am also occasionally forced to use 90 degree shutter angle as I’m always shooting at 60p (not knowing whether I’ll slow it down, which means I shoot at 1/120 instead of 1/60) and display at 30p, this means that even with a VND, I’ll still shoot between 180 and 90 shutter angle.

I’ve did some videos and no one ever notices what shutter angle I use (neither did I until I started learning more, and even now, I wouldn’t be able to tell without direct comparisons). I also shoot majority of my videos during the golden hour, which means half of my videos achieves the 180 shutter angle.

Is it still necessary for me to get a variable ND filter?
The "180 shutter speed rule" you mention is only about setting your shutter speed at about double your camera's frame rate. It is not about actually shooting with a 180 shutter speed no matter what.
So, if I shoot at 25 fps, I'll set my shutter speed at 50 (because I like my footage to look more natural -- without the strobe-like effect that comes with high shutter speeds). If I were to shoot slow motion though, I would opt to shoot at, say, 50fps and set my shutter speed at 100 (to get crisper slow motion footage), and so on and so forth.
There is not relation between the "golden hour" and shutter speed. You can shoot during that time of day, or any time of day, with whatever shutter speed you pick, provided there is enough light.
ND filters are really used to control depth of field when using a wide iris/aperture on your lens in bright conditions (like an f2.8 or wider in the middle of the day).
This is a good, clear break down of shutter speed use in video: