The upright form can easily be based from the “2”.
The italic one is a bit tricky.
I don't think you can avoid creating your own character here or putting more graphical effort in it.
If you had that glyph in some kind of digital form you could include it in your document (License!).
Code
\documentclass{scrartcl}
\usepackage[ngerman]{babel}
\usepackage{graphicx,tipa}
\newcommand*\Esh{\scalebox{-1}[1]{\rotatebox[origin=c]{180}{2}}}
\newcommand*\Eshit{\raisebox{1.75ex}{\kern-.15em\rotatebox{-20}{\scalebox{1}[-1]{2}}}\kern-.1em}
\begin{document}
Si m"ue"sti d t"a\textesh\textesh en uf \Esh tans \textesh icke \par
\emph{Si m"ue"sti d t"a\textesh\textesh en uf \Eshit tans \textesh icke} \par
\end{document}
Output

"h, "H and "Y
You can declare further short hands for other characters, too
\documentclass{scrartcl}
\usepackage[ngerman]{babel}
\usepackage{graphicx,tipa}
\newcommand*\Esh{\scalebox{-1}[1]{\rotatebox[origin=c]{180}{2}}}
\newcommand*\Eshit{\raisebox{1.75ex}{\kern-.15em\rotatebox{-20}{\scalebox{1}[-1]{2}}}\kern-.1em}
\makeatletter
\declare@shorthand{ngerman}{"h}{\textormath{\textesh}{?esh?}}
\declare@shorthand{ngerman}{"H}{\textormath{\Esh}{?Esh?}}
\declare@shorthand{ngerman}{"Y}{\textormath{\Eshit}{?Esh?}}
\makeatother
\begin{document}
Si m"ue"sti d t"a"h"hen uf "Htans "hicke \par
\emph{Si m"ue"sti d t"a"h"hen uf "Ytans "hicke} \par
\end{document}
inputenc, utf8 and newunciodechar
Make sure your TeX editor saves in UTF8.
\documentclass{scrartcl}
\usepackage[utf8]{inputenc}
\usepackage[ngerman]{babel}
\usepackage{newunicodechar}
\usepackage{graphicx,tipa}
\newcommand*\Esh{\scalebox{-1}[1]{\rotatebox[origin=c]{180}{2}}}
\newcommand*\Eshit{\raisebox{1.75ex}{\kern-.15em\rotatebox{-20}{\scalebox{1}[-1]{2}}}\kern-.1em}
\makeatletter
\declare@shorthand{ngerman}{"h}{\textormath{\textesh}{?esh?}} % still work
\declare@shorthand{ngerman}{"H}{\textormath{\Esh}{?Esh?}} % still work
\declare@shorthand{ngerman}{"Y}{\textormath{\Eshit}{?Esh?}} % still work
\makeatother
\newunicodechar{ʃ}{\textesh}
\newunicodechar{Ʃ}{\Esh}
\begin{document}
Si müeßti d täʃʃen uf Ʃtans ʃicke \par
\emph{Si müeßti d täʃʃen uf "Ytans ʃicke} \par
"h"H\textit{"Y}
\end{document}
XeTeX, LuaLaTeX
If you often find yourself writing text with many “special” characters outside of a-zA-ZäöüÄÖÜß take a look at XeTeX and LuaLaTeX.
They offer default UTF8 input and easier Unicode support. (This still doesn't help you if you can't find a font that has your special Esh.)
\rotatebox[origin=c]{180}{2}instead of\scalebox. But a font that has this special character (What kind of S is that? Is it in Unicode?) would be better. Should\texteshbe a long s (= „ſ“)? – Qrrbrbirlbel Oct 26 '12 at 18:46