I know how to do this with photoshop, but is there an easy way in illustrator to just delete this specific(marked in blue) line of pixels, while in pixel preview?
Not sure what version of illustrator your using, and there are multiple ways to handle this, but one way is…
One Simple Solve:
Go to illustrators top menu, and select view, scroll down and select “Snap to Pixel”, then drag only the offending edge/points a tiny bit one way or the other and it will snap to the pixel grid fixing your issue.
Furthermore, if you know you are creating something you want pixel perfect, there are some additional things you might do before creating your art work in illustrator. If you look in the options menu of your transform palette, you can select the “Align new objects to pixel grid” option. This combined with “Snap to Pixel”(mentioned above), will keep you from having to adjust anything at the end of your process.
When “Align new objects to pixel grid” is selected you can get some horrendous distortions to your illustrations when scaling objects not intended or designed for pixels, so if at anytime you scale something in illustrator it seems to distort, check and make sure that setting is set appropriately.
Hope this helps.