Section 7.3 of the memoir manual describes how to make your own pagestyles. (Your description of the plain style as not having a page number doesn't seem to match memoir's default, so maybe there is something else going on there.) But here's example of how you can make a pagestyle with numbers on the right:
\documentclass{memoir}
\usepackage{lipsum} % just for sample text
% Define a new pagestyle called 'right'
\makepagestyle{right}
\makeevenfoot{right}{}{}{\thepage}
\makeoddfoot{right}{}{}{\thepage}
% Make a pagestyle called 'myplain', if 'plain' has already been redefined
\makepagestyle{myplain}
\makeevenfoot{myplain}{}{\thepage}{}
\makeoddfoot{myplain}{}{\thepage}{}
\begin{document}
\pagestyle{plain} % or {myplain} if plain doesn't give what you want here
\chapter{A chapter}
\lipsum[1-10]
\pagestyle{right}
\lipsum
\end{document}
It's possible that the template you are using has redefined the plain pagestyle. If this is the case, then you can use the pagestyle myplain as defined above instead.