I have a menu type "k2 categories", that will list all articles in a certain category.

Now - This shows

<Article title>
<Intro text>
<Extra fields>
<Comments>

And so on..

Can I overwrite this template somehow, so the lookof THIS and only THIS particular category will look different, and if so - What template should I alter, and copy to where?

If I cant change only this category list, what template should alter then, so I can move <extra fields> up before <intro text> ?

Yes you can override this category only.
Create an override or template getk2.org/documentation/tutorials/174-templating-with-k2-and-the-concepts-of-sub-templates
and apply it to only that category.
Edit category.php, category_item.php, and item.php to get the desired result
you can also use css4K2 if you want it styled differently for just one category