The abstract package documentation shows the standard way of doing this without using the package:
\makeatletter
\twocolumn[
\begin{@twocolumnfalse}
\maketitle
\begin{abstract}
This is a two column abstract. It will stretch over two columns.
\end{abstract}
\end{@twocolumnfalse}
]
\makeatother
It's not clear you gain much from not using the package, though.
Many document classes store standard section names such as Abstract and References in variables with names such as \abstractname and \refname. Often you just have to change their definition.
\renewcommand\abstractname{\textbf{Abstract}}
However, in this case things are slightly more complicated, because amsart.cls uses
\scshape\abstractname
to typeset the heading for the abstract. I think it's true to say that in the default font, there are no bold small capitals, so in
\scshape\textbf{\abstractname}
the \scshape gets ignored, and the above \renewcommand works as intended. However, if you load a font that does have these glyphs (e.g. with \usepackage{mathptmx}) then 'Abstract' will end up capitalised. To prevent this, you can use the etoolbox package to get rid of \scshape entirely, and replace it with \textbf.
\documentclass{amsart}
\usepackage{etoolbox}
\usepackage{mathptmx}
\patchcmd{\abstract}{\scshape\abstractname}{\textbf{\abstractname}}{}{}
\begin{document}
\author{A Mathematician}
\title{A nice paper}
\begin{abstract}
This is the abstract.
\end{abstract}
\maketitle
\end{document}
Best Answer
The model document has
Just doing
will remove the italics.