Change position of node inside rectangle

Not exactly the same syntax, but you could potentially use a label instead of the node itself. Whether this is a viable method depends on your actual use though.

\documentclass[12pt,a4paper]{article}
\usepackage[utf8]{inputenc}
%\usepackage[german]{babel}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
\node at (0,0) [draw,minimum height=1cm,minimum width=4cm,
       label={[above,inner sep=0pt]below:Lorem ipsum}] {};

\draw (-2,-2) rectangle (2,-1);
\node at (0,-1.8) {Lorem Ipsum};
\end{tikzpicture}
\end{document}

The closest to what you like to obtain with one node:

\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
\node at (0,0) [draw,
                text height=1cm, text depth=1ex, 
                minimum width=4cm, inner sep=-0.2ex] (a) {Lorem Ipsum};

\draw (-2,-2) rectangle (2,-1);
\node at (0,-1.8) {Lorem Ipsum};
\end{tikzpicture}
\end{document}

The node above is slightly taller.

You can draw it in two steps, first the box and then position the text above the lower edge of the box.

\documentclass[12pt,a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage[german]{babel}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
  % \node at (0,0) [draw,minimum height=1cm,minimum width=4cm] {Lorem Ipsum};
  \node (Square) at (0,0) [draw,minimum height=1cm,minimum width=4cm] {};
  \node[above=-2pt] at (Square.south) {Lorem Ipsum};

  \draw (-2,-2) rectangle (2,-1);
  \node at (0,-1.8) {Lorem Ipsum};
\end{tikzpicture}
\end{document}