Table next to figure with table caption on top but figure caption on bottom

Set the tabular to be aligned at the [b]aseline as well:

\documentclass{article}

\usepackage[
  font = small,
  labelfont = bf,
  tableposition = top
]{caption}

\begin{document}

\noindent
\begin{minipage}{\textwidth}
  \begin{minipage}[b]{0.49\textwidth}
    \centering
    \rule{5.5cm}{2.6cm} % Height changed from 3.6cm to 2.6cm
    \captionof{figure}{A table beside a figure}
  \end{minipage}
  \hfill
  \begin{minipage}[b]{0.49\textwidth}
    \centering
    \captionof{table}{A table beside a figure}
    \begin{tabular}[b]{cc}
      \hline
      Table head & Table head   \\
      \hline
      Some values & Some values \\
      Some values & Some values \\
      Some values & Some values \\
      Some values & Some values \\
      Some values & Some values \\
      Some values & Some values \\
      \hline
    \end{tabular}
  \end{minipage}
\end{minipage}

\end{document}