Measure the image and use a suitably sized tabular
:
\documentclass[a4paper,12pt]{scrartcl}
\usepackage[ngerman]{babel}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{rotating}
\newsavebox{\imagebox}
\begin{document}
\begin{sidewaysfigure}
\centering
\caption{Fill}\label{fig:fill}
\sbox{\imagebox}{\includegraphics[width=0.9\paperwidth]{example-image-c}}
\begin{tabular}{@{}p{\wd\imagebox}@{}}
\usebox{\imagebox} \\
\footnotesize A long footnote, long footnote, long footnote,
long footnote, long footnote, long footnote, long footnote,
long footnote, long footnote, long footnote, long footnote,
long footnote, long footnote
\end{tabular}
\end{sidewaysfigure}
\end{document}

Note that the \label
refers to \caption
, so it's best placed near it.
A different realization with the caption left aligned with respect to the image:
\documentclass[a4paper,12pt]{scrartcl}
\usepackage[ngerman]{babel}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{rotating}
\usepackage{ragged2e}
\usepackage[
format=plain,
justification=RaggedRight,
singlelinecheck=false
]{caption}
\newsavebox{\imagebox}
\begin{document}
\begin{sidewaysfigure}
\centering
\sbox{\imagebox}{\includegraphics[width=0.9\paperwidth]{example-image-c}}
\begin{minipage}{\wd\imagebox}
\caption{Fill}\label{fig:fill}
\usebox{\imagebox}
\footnotesize A long footnote, long footnote, long footnote,
long footnote, long footnote, long footnote, long footnote,
long footnote, long footnote, long footnote, long footnote,
long footnote, long footnote
\end{minipage}
\end{sidewaysfigure}
\end{document}
