4K has four times as many pixels as 1080p so there will be absolutely no situation at all where 1080p will look better.
Just think of it, for every pixel you have at 1920 x 1080, with 4K that pixel will exist out of four pixels.
Yeah, higher resolutions do wonders. I remember back when I had a 1280x1024 monitor I absolutely needed AA, or the jaggies would murder my eyes. Now with a 1080p monitor I usually turn it off, at that reso+screen size they're not that bad for me personally.
The old 5:4 monitor was a 19 incher, and my 1080p is 21.1 inches, not *that* much bigger but the resolution difference is huge.