Align picture to the right

In order to align the image to the right margin, you might want to use foatrow's \floatsetup[figure]{margins=raggedleft} command. You can either use it globally by putting it into the preamble or locally as shown in teh following MWE:

\documentclass[12pt,a4paper]{article}
\usepackage[showframe]{geometry}
\usepackage[demo]{graphicx}
\usepackage{lipsum}
\usepackage{floatrow}
\usepackage{caption}
\captionsetup{labelsep=newline,
    justification=raggedleft,
    singlelinecheck=false,
    labelfont=bf,
    font=small}

\begin{document}
\lipsum[2]
{\floatsetup[figure]{margins=raggedleft}
\begin{figure}[H]
        \floatbox[{\capbeside\thisfloatsetup{capbesideposition={left,center},capbesidewidth=4cm}}]{figure}[\FBwidth]
        {\caption{Some caption that spans more than a line and some additional text}}
        {\includegraphics[width=5cm]{name}}
\end{figure}}

\begin{figure}[H]
        \floatbox[{\capbeside\thisfloatsetup{capbesideposition={left,center},capbesidewidth=4cm}}]{figure}[\FBwidth]
        {\caption{Some caption that spans more than a line and some additional text}}
        {\includegraphics[width=5cm]{name}}
\end{figure}

\end{document}

---

Regarding switching from article to scrartcl you will recieve the following error message:

Package caption Error: The option labelsep=newline does not work with \setcaphanging (which is set by default).

Looking through the caption manual, we can find that the default format of article and scrartcl differ:

By adding format=plain to the \captionsetup command, we can make the code compilable for scrartcl as well. Nevertheless, I'd suggest to take a look at the KOMA-script documentation as these document classes already offer ways to customize the look of captions.

Only as a supplement to the answer of @leandris for the KOMA-Script part: The KOMA-Script class provides environments captionbeside and captionofbeside.

\documentclass[12pt,a4paper]{scrartcl}
\usepackage{showframe}
\usepackage{graphicx}
\usepackage{lipsum}
\usepackage{float}% if position H is really needed

\KOMAoption{captions}{nooneline,centeredbeside,leftbeside}
\setcapindent*{0pt}
\setcaptionalignment{r}
\renewcommand\captionformat{}
\addtokomafont{caption}{\small}
\addtokomafont{captionlabel}{\bfseries}

\begin{document}
\lipsum[1]

\begin{figure}[H]
  \begin{captionbeside}
      {Some caption that spans more than a line and some additional text}%
    \includegraphics[width=5cm]{example-image}
  \end{captionbeside}
\end{figure}

\lipsum[2]

\begin{figure}[H]
  \begin{captionbeside}
      {Some caption that spans more than a line and some additional text}%
      [l]% caption position
      [\dimexpr9cm+10pt\relax]% width of caption and figure
      [\dimexpr\linewidth-9cm-10pt\relax]% offset
    \includegraphics[width=5cm]{example-image}
  \end{captionbeside}
\end{figure}

\end{document}

Result:

Warning: Package floatrow breaks environment captionbeside.