Using \hyphenation and umlauts

You need to enable an encoding that has the accented letters, then that word appears to be hyphenated as you wish already using the default hyphenation patterns.

\documentclass[a5paper]{scrartcl}
\usepackage[T1]{fontenc}
\usepackage{ngerman}

%\hyphenation{An-fangs-n"a-he-rung}

\begin{document}


\parbox{2pt}{\hspace*{1pt}Anfangsn"aherung richtig}
\end{document}

With the following MWE it should work. On my current MikTeX 2.9 it compiles without an error.

I used package babel with option ngerman and added the font encoding and input encoding. As you can see can you now write umlauts in the LaTeX document without shortcuts like \"a or "a.

\documentclass[ngerman]{scrartcl}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{babel}
\hyphenation{An-fangs-nä-he-rung}
\begin{document}
Es soll das Wort namens Anfangsnäherung richtig getrennt werden. 
\end{document}

Because package ngerman is rather old (1998, see documentation on CTAN) you should better use the maintained package babel with option ngerman (latest document version 2012).

Please have a look on english l2tabu.pdf or german l2tabu.pdf (the german version is newer) for outdated and obsolete packages you should not use longer (but you will still find them in older LaTeX documents ...).