I want page 13 to show up as 12+1

Define a new counter format:

\documentclass{article}

\usepackage{kantlipsum} % for mock text

\makeatletter
\newcommand{\fixedarabic}[1]{%
  \expandafter\@fixedarabic\csname c@#1\endcsname
}
\newcommand{\@fixedarabic}[1]{%
  \ifnum#1=13 12${}+{}$1\else\number#1\fi
}
\makeatother

\pagenumbering{fixedarabic}

\begin{document}

\kant[1-50]

\end{document}

It could be a complex solution to convert the counter page to "12+1" if the value reach 13, but if only matter when 13 is the last page, why not simply change the footer just before \end{document} to "12+1"?.

\documentclass{article}
\usepackage{fancyhdr}
\pagestyle{fancy}
\begin{document}
aaa 
\setcounter{page}{12} % this is the 12th page
\newpage
bbb
\cfoot{12+1} % this is the 12+1th page 
\end{document}

The following solution sets page 13 as 12+1 in whichever format your page numbering is set to. It uses fancyhdr to set the page number in the Centre of the footer.

\documentclass{article}

\usepackage{lipsum,fancyhdr}

\pagestyle{fancy}
\fancyhf{}% Clear header/footer
\renewcommand{\headrulewidth}{0pt}% Remove header rule
\fancyfoot[C]{%
  \ifnum\value{page}=13
   $\mbox{\setcounter{page}{12}\thepage} + \mbox{\setcounter{page}{1}\thepage}$%
   \setcounter{page}{13}%
  \else
    \thepage
  \fi
}

\pagenumbering{roman} % Just as an example

\begin{document}

\sloppy\lipsum[1-50]\lipsum[1-50]

\end{document}