Since the points may change on an \item
-by-\item
basis, insert the points manually via \insertpts{<num>}
below:

\documentclass{article}
\usepackage{enumitem}
\NewDocumentCommand{\insertpts}{ m }{%
\makebox[0pt][r]{\small\bfseries #1 pts\hspace{3em}}%
\ignorespaces}
\begin{document}
\noindent \textbf{Problem 1}
\begin{enumerate}[label=(\alph*)]
\item \insertpts{9}
Random Text
\item \insertpts{8}
Random Text
\item \insertpts{7}
Random Text
\end{enumerate}
\end{document}
You can adjust the font specifications (currently set to \small\bfseries
) and the gap by adjusting the elements inside \hspace
(currently set to 3em
).