How to force abstract and introduction to stay in the same page in LaTeX?

Unfortunately the code you showed in the question does not compile, so I'll assume some generic definition for the abstract environment.

You can avoid the automatic page break at the start of a new chapter by temporarily redefining cleardoublepage:

documentclass[12pt,a4paper,oneside]{book}
usepackage[Conny]{fncychap}

newenvironment{abstract}{}{par}

begin{document}
%microtypesetup{activate=true}
%allowhyphens
counterwithin*{subsection}{chapter}
frontmatter
tableofcontents
mainmatter
pagenumbering{arabic}
begingroup
renewcommand{cleardoublepage}{relax}
begin{abstract}
some text
addcontentsline{toc}{chapter}{Abstract}
end{abstract}
%cleardoublepage
chapter*{Introduction}
addcontentsline{toc}{chapter}{Introduction}
endgroup
end{document}