Flush a left-flushed box right

Use a tabular for the left alignment and use \hfill to flush it to the right.

\documentclass{article} 
\begin{document}

Before

\null\hfill\begin{tabular}{l@{}}
     Name \\
     Street \# \\
     City \\
     ZIP \\
 \end{tabular}

\noindent\hrule% Show width of line
After

\end{document}