How to automatically use typewriter font in arrays?

Something like this?

\documentclass{article}
\usepackage{array} % for '\newcolumntype' macro
\newcolumntype{T}{>{\ttfamily}c} 

\begin{document}
\begin{center}
\begin{tabular}{ T | T }
p  & ($\lnot$p) \\ 
\hline
1  & 0 \\
0  & 1 \\
\end{tabular}
\end{center}
\end{document}

Since \ttfamily is a text-mode command, I suggest employing a tabular environment rather than an array environment.

I think you might have an XY-problem here.

With many OpenType fonts, you would be able to use the command

\addfontfeatures{Numbers={Monospaced,Lining}}

to get fixed-width numbers in a more-attractive font. You might wrap this in an environment, such as:

\newenvironment{arraymono}%
  {\begingroup\addfontfeatures{Numbers={Monospaced,Lining}}\begin{array}}%
  {\end{array}\endgroup}

You might also find the Unicode figure space, which is exactly as wide as a tabular digit, useful here:

\newcommand\figspace{^^^^2007}