There are several errors when trying your code; actually the same error repeated six times:
! LaTeX Error: Something's wrong--perhaps a missing \item.
See the LaTeX manual or LaTeX Companion for explanation.
Type H <return> for immediate help.
...
l.33 \end
{tblr}
?
Using a minipage would help, but you also need the counter library.
\documentclass[oneside, 12pt]{book}
\usepackage{xcolor}
\usepackage{tabularray}
\UseTblrLibrary{counter}
\usepackage[export]{adjustbox}% also loads graphicx
\newtheorem{theorem}{Theorem}[section]
\begin{document}
\begin{theorem}
For every integer $k$, there exists an integer $f(k)$ such that,
for all $n$ and any $k$-arc colouring of an $n$-vertex tournament,
there is an absorbing set of size $f(k)$.
\end{theorem}
\noindent
\begin{tblr}{
colspec = {X[1,l,t]X[.5,c,t]},
stretch = 0,
rowsep = 2pt,
colsep = 0pt,
hlines = {black, 0pt},
vlines = {black, 0pt},
}
\begin{minipage}[t]{\dimeval{2\textwidth/3}}\begin{theorem}
If a $k$-arc-coloured tournament $T$ has no rainbow triplet
(that is, a triplet of vertices such that all 3 arcs have
different colours), then $T$ has an absorbing vertex.
\end{theorem}\end{minipage}
&
\quad\includegraphics[width=\dimeval{\textwidth/3-1em},valign=t]{example-image}
\end{tblr}
\end{document}

The same output can be obtained without tblr:
\documentclass[oneside, 12pt]{book}
\usepackage{xcolor}
\usepackage{tabularray}
\UseTblrLibrary{counter}
\usepackage[export]{adjustbox}% also loads graphicx
\newtheorem{theorem}{Theorem}[section]
\begin{document}
\begin{theorem}
For every integer $k$, there exists an integer $f(k)$ such that,
for all $n$ and any $k$-arc colouring of an $n$-vertex tournament,
there is an absorbing set of size $f(k)$.
\end{theorem}
\noindent
\begin{minipage}[t]{\dimeval{2\textwidth/3}}\begin{theorem}
If a $k$-arc-coloured tournament $T$ has no rainbow triplet
(that is, a triplet of vertices such that all 3 arcs have
different colours), then $T$ has an absorbing vertex.
\end{theorem}\end{minipage}\quad
\includegraphics[width=\dimeval{\textwidth/3-1em},valign=t]{example-image}
\end{document}
\newtheorem{corollary}{Corollary}[theorem]should be\newtheorem{corollary}[theorem]{Corollary}, note I would to the same with all the others that are not thetheoremom. Then they all count using the same counter, which is better for the reader. You only need[section]on the one that definedstheorem