Is it possible to refer to the last column of a tikz matrix?

pgf has the counts \pgfmatrixcurrentrow and \pgfmatrixcurrentcolumn, which get reset whenever you start a new matrix. So if you inspect the counts right after a matrix, it they will contain the number of rows and columns. Otherwise you can store them in macros. However, in your example you need only

\documentclass[tikz,border=3mm]{standalone}

\usetikzlibrary{matrix}

\begin{document}
\begin{tikzpicture}
  \matrix[
  , matrix of nodes
  , left delimiter={[},right delimiter = {]}
  ] (m)
  {
    * & * & * & * \\
    * & * & * & * \\
    * & * & * & * \\
  };

  \node at (m-2-\the\pgfmatrixcurrentcolumn) {\textbullet};

\end{tikzpicture}
\end{document}

If the number of columns is smaller than the maximum in the last row, the above method fails. You can define styles for this case. Starting from pgf version 3.1.6 there is a method that allows you to smuggle the results out of the path. You can then retrieve them after using some appropriate pop.

\documentclass[tikz,border=3mm]{standalone}
\usetikzlibrary{matrix}
\makeatletter
\tikzset{record number of columns in/.style={execute at end matrix={%
\edef#1{\the\pgf@matrix@numberofcolumns}%
\pgfutil@pushmacro#1}},
record number of rows in/.style={execute at end matrix={%
\edef#1{\the\pgfmatrixcurrentrow}%
\pgfutil@pushmacro#1}}
}
\newcommand\pgfpop[1]{\pgfutil@popmacro#1}
\makeatother
\begin{document}
\begin{tikzpicture}
  \matrix[matrix of nodes,
  left delimiter={[},right delimiter = {]},
  record number of columns in=\mycols,
  record number of rows in=\myrows
  ] (m)
  {
    * & * & * & * \\
    * & * & * & * \\
    * & * & * & * \\
  };
  \pgfpop\mycols
  \pgfpop\myrows
  \node[anchor=center] at (m-2-\mycols.center) {\textbullet};
\end{tikzpicture}
\end{document}

Alternatively you can introduce new counts.

\documentclass[tikz,border=3mm]{standalone}
\usetikzlibrary{matrix}
\newcount\tikzmatrixrows
\newcount\tikzmatrixcols
\makeatletter
\tikzset{record matrix dimensions/.style={execute at end matrix={%
\global\tikzmatrixcols=\pgf@matrix@numberofcolumns
\global\tikzmatrixrows=\pgfmatrixcurrentrow}}}
\makeatother
\begin{document}
\begin{tikzpicture}
  \matrix[matrix of nodes,
  left delimiter={[},right delimiter = {]},
  record matrix dimensions
  ] (m)
  {
    * & * & * & * \\
    * & * & * & * \\
    * & * & * & * \\
  };
  \node[anchor=center] at (m-2-\the\tikzmatrixcols.center) {\textbullet};
\end{tikzpicture}
\end{document}

Finally, you do not need to know the number explicitly.

\documentclass[tikz,border=3mm]{standalone}
\usetikzlibrary{matrix}
\begin{document}
\begin{tikzpicture}
  \matrix[matrix of nodes,nodes={alias=m-\the\pgfmatrixcurrentrow-last},
  left delimiter={[},right delimiter = {]}
  ] (m)
  {
    * & * & * & * \\
    * & * & * & * \\
    * & * & * & * \\
  };

  \node at (m-2-last) {\textbullet};

\end{tikzpicture}
\end{document}