I've cropped this picture with the lasso tool:
What's my next move about this? Do I do this manually by removing pixel by pixel or are there some quicker solutions for this task?
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First you want to start with the best selection possible. Here you have some choices.
Near the bottom you'll see Decontaminate Colors. This will remove some of the blue sheen that may be on the edge of the subject from the background sky. This refinement will do a pretty good job on solid edges and a fair job on hair.
After making your selection, I would use it to create a mask. Then refine the mask. Using a mask means you don't delete pixels from your original, so it's non-destructive. Again, several choices:
In the masks panel, select Mask Edge. This will give you some sliders to work with. I'd suggest selecing "Smart Radius" and increase the radius to a few pixels. This is the same tool as Refine Edge from above, that you may have used along with the quick select.
Or go into quick mask (shortcut is "Q" key) and then you can use a soft white or black soft brush to tidy up areas. I wouldn't do pixel by pixel, just use a small soft brush very carefully.
You could also get a good mask by the following steps: