There is a way to do this, but it's not very easy to do. If memory serves, each report style in Service Desk corresponds to a xsl file in the client\repo\xsl\form directory and you can control which elements are displayed by modifying the xsl. You may want to start by remove specific elements and knowng a little HTML will definitely be helpful.
At one point, I set out to create a custom print report style but we found that a better way to control printing is to make a small web application (linked through a smart action) that accepts the item number as an input through the URL and then uses the item number to display specific fields from the item in a browser. Calling a third party application from Service Desk gives you more control over of the formatting but maybe removing some of the items in the report style will work well enough for your customers.