Enumerate alignment

If I understand correctly, you need to add [t] alignment:

\documentclass{article}

\begin{document}

\begin{enumerate}
\item  \begin{tabular}[t]{cccccccccc}        %%%% note [t] here.
   $x[n] = $ & \{1/3 & -1/2 & 1/4 & -1/8 & 1/6\} & and\\
      &   &   & $\uparrow$   &   &  &  \\
  \end{tabular}
\begin{tabular}[t]{cccccccccc}               %%%% and here.
   $h[n] = $ & \{1 & -1 & 1 & -1\} \\
             &    &   &    &   &    &    \\
  \end{tabular}
\item \begin{tabular}{cccccccccc}
   $x[n] = $ & \{1 & -2 & -3 & 0 & -1\} & and\\
  \end{tabular}
\begin{tabular}{cccccccccc}
   $h[n] = $ & \{2 & -1/2 & -3 & 1 & -2\} \\
  \end{tabular}
\end{enumerate}

\end{document}

The following is a suggestion for a slightly easier notation that is more flexible. Moreover, it provides a consistent and accurate spacing around the math operators you use:

\documentclass{article}
\usepackage{amsmath,xparse}% http://ctan.org/pkg/{amsmath,xparse}
\NewDocumentCommand\printarray{O{~~} >{\SplitList{,}}m}
{%
  \def\itemdelim{\def\itemdelim{#1}}% Define list separator with one delay
  \ProcessList{#2}{\myitem}% Process list
}
\newcommand\myitem[1]{\itemdelim{#1}}
\begin{document}

\begin{enumerate}
  \item $x[n] = \{\printarray{1/3,-1/2,\underset{\uparrow}{1/4},-1/8,1/6}\}$ and $h[n] = \{\printarray{1,-1,1,-1}\}$
  \item $x[n] = \{\printarray{1,-2,-3,0,-1}\}$ and $h[n] = \{\printarray{2,-1/2,-3,1,-2}\}$
\end{enumerate}

\end{document}

The "array" is printed using \printarray[<sep>]{<CSV list>} where <sep> defaults to ~~.

Delayed definition of the list separator stems from Package xparse \SplitList last token (or originally Cunning (La)TeX tricks), while amsmath provides \underset.

I also would like to suggest you another approach, using a matrix environment instead of a tabular; it's more natural, gives you the desired vertical alignment automatically, and saves you the column format specification:

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\begin{enumerate}
\item 
$
x[n] = 
\{\begin{matrix} 
1/3 & -1/2 & \smash{\underset{\uparrow}{1/4}} & -1/8 & 1/6
\end{matrix}\} 
$\quad
and\quad 
$
h[n] =
\{\begin{matrix} 
1 & -1 & 1 & -1
\end{matrix}\}
$
\item 
$
x[n] = 
\{\begin{matrix}
1 & -2 & -3 & 0 & -1
\end{matrix}\}
$\quad
and\quad 
$
h[n] =
\{\begin{matrix}
2 & -1/2 & -3 & 1 & -2
\end{matrix}\}
$
\end{enumerate}

\end{document}