The caption package is loaded by the apa6 class, so you can use its commands to set the caption formatting. Adding:
\captionsetup{justification=centering}
will cause the caption to be centred. However, this doesn't seem to be the exact formatting you want. What you want a left aligned caption aligned to the natural width of the table. As Axel Sommerfeldt notes in the comments, there is an command in the caption package which allows you to place a table and its caption in a box. This will allow you to do what you want quite simply. The command has not yet made it into the documentation of caption (v3.2).
The syntax of this command is:
\captionbox[⟨list entry⟩]{⟨heading⟩}[⟨width⟩][⟨inner-pos⟩]{⟨contents⟩}
\captionbox*{⟨heading⟩}[⟨width⟩][⟨inner-pos⟩]{⟨contents⟩}
(quoting from the subcaption package by the same author:)
The arguments ⟨list entry⟩ & ⟨heading⟩ will be used for
typesetting the \caption. ⟨width⟩ is the width of the resulting
\parbox; the default value is the width of the contents.
⟨inner-pos⟩ specifies how the contents will be justified inside the
resulting \parbox; it can be either c (for \centering), l (for
\raggedright), r (for \raggedleft), or s (for no special
justification). The default is c. (But you can use any justification
defined with \DeclareCaptionJustification as well, e.g.
centerlast.)
\documentclass[doc]{apa6}
\usepackage[american,ngerman]{babel}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\shorttitle{A title}
\begin{document}
Example of a table:
\begin{table}[htbp]
\centering
\captionbox{Some caption\label{table1}}{%
\begin{tabular}{lcc}
\toprule
Test1 & 1 & 2\\
Test2 & 1 & 2\\
Test3 & 1 & 2\\
\bottomrule
\end{tabular}
}
\end{table}
\end{document}

Unless you have very strict formatting requirements, I would simply stick to centred captions instead of doing this.
(Thanks to both Werner and Axel for their comments on this answer.)
apa6class requires\shorttitleto be used, so you can't just piece together your fragments. Also, you should use\centeringnot\centerinside your tables. – Alan Munn Jan 25 '12 at 3:00apa6document classs: Is this a requirement of your school or is this something you can select yourself? A design choice embedded in theapa6eclass is that all float captions are typeset left-justified on two lines, with a line break between "Tabelle x" (in German because of your babel setting) and the actual text of the caption. In addition, the tabular materials should also be set left-justified, i.e., you shouldn't use the\centeringcommand. If you don't like these design choices, a different document class may be better suited to your needs. – Mico Jan 25 '12 at 3:17