Here is a solution :
\documentclass[a4paper, 12pt, oneside]{memoir}
\usepackage[english]{babel}
\setcounter{page}{45}
\usepackage{lipsum}
\let\oldchapter\chapter
\makeatletter
\def\chapter{%
\@ifstar{\@Starred}{\@nonStarred}%
}
\def\@Starred{%
\@ifnextchar[%
{\GenericWarning{}{Warning: A starred chapter can not have parameters. I am going to ignore them!}\@StarredWith}%
{\@StarredWithout}%
}
\def\@StarredWith[#1]#2{%
\oldchapter{#2}%
\makeoddhead{headings}{\slshape#1}{}{\thepage}%
}
\def\@StarredWithout#1{%
\oldchapter*{#1}%
\makeoddhead{headings}{\slshape\rightmark}{}{\thepage}%
}
\def\@nonStarred{%
\@ifnextchar[%
{\@nonStarredWith}%
{\@nonStarredWithout}%
}
\def\@nonStarredWith[#1]#2{%
\oldchapter[#1]{#2}%
\makeoddhead{headings}{\slshape\rightmark}{}{\thepage}%
}
\def\@nonStarredWithout#1{%
\oldchapter{#1}%
\makeoddhead{headings}{\slshape\rightmark}{}{\thepage}%
}
\makeatother
\begin{document}
\tableofcontents
\chapter*[My short chapter title]{My Very Long Chapter Title Which Is Hated}
\newpage
Some content
\chapter{one other chapter here}
\lipsum[2]
\chapter*[ToC title for chapter 3]{Chapter3 full title}
\section{Test section}
\lipsum[3]
\end{document}
I have used my redefinition of \section (but used it for \chapter) from here: https://tex.stackexchange.com/a/380116/120578
The idea is to use:
\chapter*[short title for header]{title for chapter and TOC}
Starred chapter with optional does not ordinary exist but can be used with the code above.
Output of TOC:

Output of chapter1 title:

Output of next page header:

Edit:
Another option that uses your default settings and breaks the title using minipage (I don't recomend it but just adding to say you can do several things as reducing the fontsize... ignoring the word "Chapter" etc):
Also added because I have used closer definitions of headers to your initial.
\documentclass[a4paper, 12pt, oneside]{memoir}
\usepackage[english]{babel}
\setcounter{page}{45}
\usepackage{lipsum}
\let\oldchapter\chapter
\makeatletter
\def\chapter{%
\@ifstar{\@Starred}{\@nonStarred}%
}
\def\@Starred{%
\@ifnextchar[%
{\GenericWarning{}{Warning: A starred chapter can not have parameters. I am going to ignore them!}\@StarredWith}%
{\@StarredWithout}%
}
\def\@StarredWith[#1]#2{%
\oldchapter{#2}%
\makeoddhead{headings}{\begin{minipage}{0.9\linewidth}\rightmark\end{minipage}}{}{\thepage}%
}
\def\@StarredWithout#1{%
\oldchapter*{#1}%
\makeoddhead{headings}{\MakeUppercase{\rightmark}}{}{\thepage}%
}
\def\@nonStarred{%
\@ifnextchar[%
{\@nonStarredWith}%
{\@nonStarredWithout}%
}
\def\@nonStarredWith[#1]#2{%
\oldchapter[#1]{#2}%
\makeoddhead{headings}{\MakeUppercase{\rightmark}}{}{\thepage}%
}
\def\@nonStarredWithout#1{%
\oldchapter{#1}%
\makeoddhead{headings}{\MakeUppercase{\rightmark}}{}{\thepage}%
}
\makeatother
\begin{document}
\tableofcontents
\chapter*[nothing needed here]{My Very Long Chapter Title Which Is Hated}
\newpage
Some content
\chapter{one other chapter here}
\lipsum[2]
\chapter[ToC title for chapter 3]{Chapter3 full title}
\newpage
Test
\section{Test section}
\lipsum[3]
\end{document}
\chapter, e.g.\chapter[My short Title]{My Very Long Chapter Title Which Is Hated}. With this,My short Titlegoes into the header. But be aware, the short title is also used in the TOC. – Mike Oct 16 '17 at 20:34\markboth{short header text}{short header text}. that leaves the full title free to be used in the toc. – barbara beeton Oct 17 '17 at 01:09