I don't get an error using tocloft. So I cannot reproduce your issue with an already defined \@starttoc.
But \cftsetpnumwidth is the wrong length. Not the page numbers are the problem, but the with of the section number. So you would have to change \cftsecnumwidth:
\documentclass{article}
\usepackage{tocloft}
\setlength{\cftsecnumwidth}{2.3em}
\usepackage[toc,title]{appendix} %% Manage appendix when using \appendices environment: toc = put header in TOC; title = add the word 'Appendix' in the document before each number
\usepackage{alphalph} %% To number appendices beyond Z
\begin{document}
\tableofcontents\pagebreak% NOTE: \pagebreak here is wrong. It should be either \clearpage or \newpage or better be removed.
\appendices
\makeatletter
\newalphalph{\Alphmult}[mult]{@Alph}{26}
\makeatother
\renewcommand{\thesection}{\Alphmult{\value{section}}}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\pagebreak % So it doesn't overrun
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{The last appendix}
\end{document}

See page 9 (section 2.3) of the tocloft manual for more information about \cftXnumwidth and the name you have to use for X.
As an alternative to using tocloft you can patch \l@section. The original code in article.cls for setting the width reserved for the section number of the ToC entry is: \setlength\@tempdima{1.5em}. So you could use:
\documentclass{article}
\usepackage[toc,title]{appendix} %% Manage appendix when using \appendices environment: toc = put header in TOC; title = add the word 'Appendix' in the document before each number
\usepackage{alphalph} %% To number appendices beyond Z
\usepackage{xpatch}
\makeatletter
\xpatchcmd{\l@section}{%
\setlength@tempdima{1.5em}%
}{%
\setlength@tempdima{2.3em}%
}{}{\PatchFailure}
\makeatother
\begin{document}
\tableofcontents
\appendices
\makeatletter
\newalphalph{\Alphmult}[mult]{@Alph}{26}
\makeatother
\renewcommand{\thesection}{\Alphmult{\value{section}}}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\pagebreak % So it doesn't overrun
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{The last appendix}
\end{document}
The result would be the same as above.
But because, usually you would also like to change the indent of the subsection and the subsubsection entries. And maybe you would also like to change the number width of these entries. So using tocloft and \setlength{\cftsubsecindent}{…}` etc. could be favorably.
Another completely different suggestion would be to use package tocbasic instead of tocloft. It provides the feature to detect the needed width of the number automatically:
\documentclass{article}
\usepackage[toc,title]{appendix} %% Manage appendix when using \appendices environment: toc = put header in TOC; title = add the word 'Appendix' in the document before each number
\usepackage{alphalph} %% To number appendices beyond Z
\usepackage{tocbasic}
\DeclareTOCStyleEntry[dynnumwidth]{tocline}{section}% automatically detect the number width
\begin{document}
\tableofcontents
\appendices
\makeatletter
\newalphalph{\Alphmult}[mult]{@Alph}{26}
\makeatother
\renewcommand{\thesection}{\Alphmult{\value{section}}}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\pagebreak % So it doesn't overrun
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{An appendix}
\section{The last appendix}
\end{document}
This needs as least three LaTeX runs to result in:

See section 15.3 of one of the KOMA-Script manuals for more information aboute \DeclareTOCStyleEntry.
There are several more packages to manipulate the ToC.