If you want to customize the tile of the references section, you can use \defbibheading:
\defbibheading{bibliography}{\centering The New Name}
With the help og the fancyhdr package you can define a new page style (with customized headers and footers) to be applied to the references section:
\documentclass[noapacite]{apa}
\usepackage[backend=biber]{biblatex}
\usepackage{fancyhdr}
\usepackage{filecontents}
\fancypagestyle{myrefstyle}{
\fancyhf{}
\fancyhead[C]{The New Header}
\fancyhead[R]{\thepage}
\renewcommand\headrulewidth{0pt}
}
\begin{filecontents}{\jobname.bib}
@thesis{A01,
author = {Megalomanius, M.},
year = {1900},
title = {Why I am so great}},
}
\end{filecontents}
\addbibresource{\jobname.bib}
\begin{document}
\nocite{*}
\clearpage
\pagestyle{myrefstyle}
\printbibliography
\end{document}

If the twoside option is in use, you'll have to replace
\fancyhead[R]{\thepage}
with
\fancyhead[RO,LE]{\thepage}
in my example code.
The apa document class uses upper case italics to format the headers, so you can use something like
\fancypagestyle{myrefstyle}{
\fancyhf{}
\fancyhead[C]{\MakeUppercase{\itshape The New Header}}
\fancyhead[RO,LE]{\thepage}\renewcommand\headrulewidth{0pt}
}
to get the same style as the one used by apa.cls.