I think ultimately the correct way to solve this problem is with textpos, but it's certainly possible to do so with tables (as in the previous answer) or with creative mucking around with TeX boxes.
If you choose the latter course, the trick is to get TeX to treat the boxes as independent for positioning purposes. I do this by making them all essentially of zero dimensions: I wrap them all in \vbox to0pt{\vss\rlap{\llap{contents}}\vss} so that TeX thinks that they're 0 points tall and 0 points wide. This lets me position them as I will. Then it's just a matter of \hskiping and \vskiping everything into place.
\documentclass{amsbook}
\usepackage{afterpage}
\usepackage{graphicx}
\usepackage{lmodern}
\usepackage{xcolor}
\definecolor{emerald}{HTML}{0a6666}
\newcommand{\upperbold}[1]{\textbf{\textsf{\MakeUppercase{#1}}}}
\begin{document}
\begin{titlepage}
\color{white}
\pagecolor{emerald}\afterpage{\nopagecolor}
\noindent\hskip-0.75in\vbox to0pt{%
\vskip2in%
\fontsize{42pt}{42pt}\selectfont%
\noindent\upperbold{Hans Hagen}\bigskip\par%
\vskip1em%
\fontsize{35pt}{35pt}\selectfont%
\noindent\upperbold{Context MKIV}\bigskip\par%
}%
\vskip3.45in%
\hskip0.75\linewidth%
\vbox to0pt{%
\vss%
\llap{%
\rlap{%
\rotatebox{-90}{%
{\fontsize{220pt}{220pt}\selectfont\upperbold{UNITS}}%
}%
}%
}%
\vss%
}%
\end{titlepage}
\end{document}
This coughs up the following:

Now, I tend to sprinkle % at the end of all my lines in the paranoid fear that I'll forget it where it's actually needed, and I did much more indenting than is really required here, but it does the trick.
Still, I'll say again: I think looking into textpos is the way to go. But since you seemed to want something more basic TeX/LaTeX, I did it that way.