Move right-aligned paragraph more to the center

You can simply specify a different \rightskip inside the flushright. flushright works by putting \rightskip to 0pt and \leftskip to exandable glue. Replace the \rightskip value by some positive length, e.g. by

\begin{flushright}%
  \setlength{\rightskip}{3em}%
  ...
\end{flushright}

\documentclass[fontsize=12pt,paper=A5,DIV=14,headinclude=true,footinclude=true,headings=small,pagesize,parskip=half]{scrartcl}

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[english,naustrian]{babel}

\usepackage{multicol, sectsty}

\raggedbottom
\addtolength{\topmargin}{-1cm}
\addtolength{\textheight}{2.5cm}
\allsectionsfont{\large\centering}

\begin{document}
    \section{The Little Drummer Boy}
        \begin{multicols}{2}

            1. Come, they told me,\\ 
            \emph{pa rum pum pum pum},\\
            our new born King to see, \\
            \emph{pa rum pum pum pum},\\
            our finest gifts we'll bring, \\
            \emph{pa rum pum pum pum},\\
            to lay before the King, \\
            \emph{pa rum pum pum pum},\\
            \emph{rum pum pum pum,\\ rum pum pum pum.}

            So to honor Him,\\ 
            \emph{pa rum pum pum pum},\\
            when we come.

        \columnbreak

            2. Little Baby, \\
            \emph{pa rum pum pum pum},\\
            I am a poor boy, too,\\
            \emph{pa rum pum pum pum},\\
            I have no gift to bring,\\
            \emph{pa rum pum pum pum},\\
            that's fit to give a King,\\
            \emph{pa rum pum pum pum},\\
            \emph{rum pum pum pum,\\ rum pum pum pum}

            Shall I play for you,\\
            \emph{pa rum pum pum pum},\\
            on my drum? 

    \end{multicols}
    \begin{flushright}
      Ordinary flushright text.
    \end{flushright}
    \begin{flushright}%
      \setlength{\rightskip}{3em}%
            3. Mary nodded, \emph{pa rum pum pum pum},\\
            the ox and lamb kept time, \emph{pa rum pum pum pum},\\
            I played my drum for Him, \emph{pa rum pum pum pum},\\
            I played my best for Him, \emph{pa rum pum pum pum},\\
            rum pum pum pum, rum pum pum pum.

            Then He smiled at me, \emph{pa rum pum pum pum},\\
            me and my drum.

    \end{flushright}
\end{document}

Here are two examples where you don't need to specify an explicit length:

\documentclass[fontsize=12pt,paper=A5,DIV=20,headinclude=true,footinclude=true,headings=small,pagesize,parskip=half]{scrartcl}

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[english,naustrian]{babel}
\usepackage[babel=true,protrusion=true,factor=900,draft=false]{microtype}

\usepackage{multicol, sectsty}

\usepackage{varwidth}

\raggedbottom
\addtolength{\topmargin}{-1cm}
\addtolength{\textheight}{2.5cm}
\allsectionsfont{\large\centering}

\begin{document}
    \section{The Little Drummer Boy}
        \begin{multicols}{2}

            1. Come, they told me,\\ 
            \emph{pa rum pum pum pum},\\
            our new born King to see, \\
            \emph{pa rum pum pum pum},\\
            our finest gifts we'll bring, \\
            \emph{pa rum pum pum pum},\\
            to lay before the King, \\
            \emph{pa rum pum pum pum},\\
            \emph{rum pum pum pum,\\ rum pum pum pum.}

            So to honor Him,\\ 
            \emph{pa rum pum pum pum},\\
            when we come.

        \columnbreak

            2. Little Baby, \\
            \emph{pa rum pum pum pum},\\
            I am a poor boy, too,\\
            \emph{pa rum pum pum pum},\\
            I have no gift to bring,\\
            \emph{pa rum pum pum pum},\\
            that's fit to give a King,\\
            \emph{pa rum pum pum pum},\\
            \emph{rum pum pum pum,\\ rum pum pum pum}

            Shall I play for you,\\
            \emph{pa rum pum pum pum},\\
            on my drum? 

    \end{multicols}

    \noindent
    \rule{\textwidth}{1pt}

    \begin{center}
      \begin{varwidth}{\linewidth}
        \begin{flushright}
          3. Mary nodded, \emph{pa rum pum pum pum},\\
          the ox and lamb kept time, \emph{pa rum pum pum pum},\\
          I played my drum for Him, \emph{pa rum pum pum pum},\\
          I played my best for Him, \emph{pa rum pum pum pum},\\
          rum pum pum pum, rum pum pum pum.

          Then He smiled at me, \emph{pa rum pum pum pum},\\
          me and my drum.
        \end{flushright}
      \end{varwidth}
    \end{center}

    \begin{center}
      \begin{tabular}{@{}r@{}}
        3. Mary nodded, \emph{pa rum pum pum pum},\\
        the ox and lamb kept time, \emph{pa rum pum pum pum},\\
        I played my drum for Him, \emph{pa rum pum pum pum},\\
        I played my best for Him, \emph{pa rum pum pum pum},\\
        rum pum pum pum, rum pum pum pum.\\
        Then He smiled at me, \emph{pa rum pum pum pum},\\
        me and my drum.
      \end{tabular}
    \end{center}
  \end{document}