Center column headings in tabular: multicolumn not respected

It is centered correctly. The problem is the length (width) of numbers in that column that are right aligned. I mean to say α = 0.99999 is longer than any of the numbers in that column. As a proof, let us put a slightly bigger number.

\documentclass{article}

\usepackage[american]{babel}
\usepackage{tabularx}
\usepackage{booktabs}

\begin{document}
\begin{tabular}{rr}
  \toprule
  \multicolumn{1}{c}{$\alpha$} & \multicolumn{1}{c}{$\alpha=0.99999$} \\
  \midrule
  10     & 69.681   \\
         & 901.741  \\
  1000   & 893.630  \\
         & 82.80600000000000000000   \\
  \bottomrule
\end{tabular}
\end{document}

This is just a little example of how you can make a small modification to your input in order to get the desired output:

\documentclass{article}
\usepackage{tabularx}% http://ctan.org/pkg/tabularx
\usepackage{booktabs}% http://ctan.org/pkg/booktabs
\begin{document}
\begin{tabular}{rc}
  \toprule
  \multicolumn{1}{c}{$\alpha$} & $\alpha=0.99999$ \\
  \midrule
  10     & \phantom{0}69.681   \\
         & 901.741  \\
  1000   & 893.630  \\
         & \phantom{0}82.806   \\
  \bottomrule
\end{tabular}
\end{document}

The addition of a \phantom leading 0 for numbers less then 100 makes all the numbers the same width, so centering is readily achieved.

Using the dcolumn package you can get the desired alignment automatically:

\documentclass{article}
\usepackage{booktabs}
\usepackage{dcolumn}

\newcolumntype{M}{D{.}{.}{3.3}}

\begin{document}

\begin{tabular}{rM}
\toprule
\multicolumn{1}{c}{$\alpha$} & \multicolumn{1}{c}{$\alpha=0.99999$} \\
\midrule
10     & 69.681   \\
       & 901.741  \\
1000   & 893.630  \\
       & 82.806   \\
\bottomrule
\end{tabular}

\end{document}