Q:

How to add print link action of specific pages?

I think the expected view (Fig. 2) is necessary because I work in environments where are many printers. 
I know the thread How to add print link answer which opens a general print window without the ability to specify fine details such as which pages to print by activation of the button. Reproduction of the code here with the addition of the target of this thread - customized action of the print button
% https://tex.stackexchange.com/a/30534/13173
\documentclass{scrbook}
\usepackage{hyperref}
\usepackage{pdfpages}
\usepackage{lipsum}

\begin{document}
\Acrobatmenu{Print}{Print doc}
\lipsum

% TODO include a link here which will print out the included pdf from the document
\includepdf[page={1-3}]{leo.pdf} % choose any multipage pdf file here 

\end{document}

Aim

to include Print button on page which will print specific pages of the document
to include Print button which will print current page of the document. 

Fig. 1 Output of AlexG's button action where you get two times a popup, 
Fig. 2 View when pressing GUI's Print button

Expected output GUI: Wanted settings in the GUI (Fig. 2)    
Testing AlexG's proposal
Output in Fig. 1. 
Engines: pdfTeX, XeLaTeX, ...
OS: Debian 8.7
PDF viewer: Adobe Acrobat 9.x (acroread) installed as described here 

A:

This requires JavaScript and perhaps A-Reader. The buttons themselves are not printed. All engines.
Basic example using print() method with parameters: No print dialog possible when specifying page ranges; multiple calls of print() method needed for multiple page ranges:
\documentclass{article}

\usepackage{media9} % non-printable push buttons with JS actions
\usepackage{pgffor}

\begin{document}

Page \thepage

% print two page ranges
% Note: nStart/nEnd are zero-based
\mediabutton[
  jsaction={
    this.print({
      bUI: false, %no print dialog; mandatory if page range is specified
      nStart: 0,
      nEnd: 3,
    });
    this.print({
      bUI: false,
      nStart: 6,
      nEnd: 8,
    });
  }
]{\fbox{Print pp. 1--4 and 7--9}}

\foreach \i in {2,...,10} {
  \newpage
  Page \thepage

  % print current page
  \mediabutton[
    jsaction={
      this.print({
        bUI: false,
        nStart: this.pageNum,
        nEnd: this.pageNum,
      })
    }
  ]{\fbox{Print current page}}
}

\end{document}

Example with advanced settings using PrintParams object (only recent AR versions on Windows and OSX): Now, settings (target printer and other parameters) can be modified in the print dialog.
Properties of the PrintParams object are documented in the JavaScript for Acrobat API Reference.
\documentclass{article}

\usepackage{media9}
\usepackage{pgffor}

\begin{document}

Page \thepage

%print two page ranges, show dialog
\mediabutton[
  jsaction={
    var pp = this.getPrintParams();
    pp.interactive = pp.constants.interactionLevel.full; %show print dialog
    %pp.interactive = pp.constants.interactionLevel.automatic; %suppress dialog, but progress monitor and "cancel" shown
    %pp.interactive = pp.constants.interactionLevel.silent; %suppress everything
    pp.printRange=[[0, 3],[6, 8]]; %two ranges, again 0-based page nums
    this.print(pp);
  }
]{\fbox{Print pp. 1--4 and 7--9}}

\foreach \i in {2,...,10} {
  \newpage
  Page \thepage

  % (silently) print current page
  \mediabutton[
    jsaction={
      var pp = this.getPrintParams();
      pp.interactive = pp.constants.interactionLevel.silent;
      pp.firstPage=this.pageNum;
      pp.lastPage=this.pageNum;
      this.print(pp);
    }
  ]{\fbox{Print current page}}
}

\end{document}

Many of the settings in the print dialog can be pre-configured via the PrintParams object properties.

