From a screenshot I want to cut only the part that is important. I do know I can do it using GIMP, but GIMP is very slow to start. Is there any image application that can do it faster?
If its something you want to do more than once (say from multiple screenshots) then its best to Alt-Print Screen to perform the image cuts and then use Imagemagick convert as follows:
convert -crop 400x300+50+200 screenshot.png test.png
will generate a 400x300 image as a subset of the original screenshot.png offset by 50 pixels east and 200 pixels south.
Its slow to sort the first time, but once you have done it once the other images can be put into a batch using a bash script.