Should be the same process in any image editing software, although the
terms may vary.

You may want to consider that size of the original capture at 640 x 400
pixels is small enough for display on most user's systems.  320 x 200
pixels looks pretty small on my monitor running at 1600 x 1200.  The
"cost" for the larger image is small - where at 320 x 200 the final file
size can be around 9kb, at 640 x 400, file size only goes up to around
<13kb - pretty good considering that the image covers 4 times the area.

Gimp is freeware - versions for windows, unix, mac, are available.
Photoshop is also fine, but outrageously priced for most things people
use it for.  I could talk you through using photoshop 7, but IMO you'd
still want to use a tool to optimise png compression, so rather than
those, I'd suggest using just Irfanview as an "all in one" solution (you
might be able to use it for screen capture as well).
http://www.irfanview.com/
Download the Irfanview plug-ins package, as you'll need the "pngout" plugin.

Open the screen capture image in irfanview, maximise the window, and
select | view | display options | Fit images to window.
Use your mouse to select a rectangle around the image. Then Edit | Crop.
 This gets rid of the black border. I didn't have too much trouble
selecting the 640x400 rectangle, but YMMV.
Resize the image down to half size (if you want to go from the original
640 x 400 to 320 x 200).  Make sure that the size is exactly half -
there's a button in the "Resize Resample" dialog to choose half size.
Also make sure that "apply sharpen after resample" is unchecked, and
that "resize (faster)" and not "resample (better quality)" is selected.
Then save as - select "Save as type" "png - portable network graphics".
 With the plugins installed, this makes an additional "pngout" dialog
appear.  Make sure that the check box "Use Pngout Plugin is selected.
Most default options are probably fine - but change the compression
level to "9". In this case the pngout plugin automatically determines
the number of unique colours in the image, and changes colour mode from
rgb to indexed (*in this case 40 unique colours).  I also unselected the
default Keep Chunks "pHYs" to "None" and don't know enough about why,
just that the PNGOut command line tool has none as default, and I've no
idea why the Irfanview plugin is different.
Hit the save button - job done.

*Alternately, in Irfanview you can select Image | Reduce colour depth,
and preview the effect of reducing the palette down further to save file
size - down to about 32 colours actually seems okay for your test image,
but by 16 colours, the gradient in the sky becomes posterised (striped).
 If you do it this way, usually make sure that the "Use Floyd-Steinberg
Dithering" is unselected.  As an example, the increased complexity of
dithering when reducing palette to 16 colours costs more in final file
size (and give IMO a worse result) than just sticking with the original
40 colours.