How can I left align a caption for a centered table?

Note the addition of @{} to each side of the tabular.

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[font=small,labelfont=bf]{caption}

\begin{document}

\begin{table}
  \sbox0{\begin{tabular}{@{}ll@{}}
         Processor & Intel(R) Core(TM) i7-7500U CPU @2.70GHz 2.90 GHz \\
         Installed RAM & 8.00 GB (7.89 GB usable)
    \end{tabular}}%
  \centering
  \begin{minipage}{\wd0}
    \captionsetup{singlelinecheck = false, justification=justified}
    \caption{System specifications}
    \label{tab:specs}
    \usebox0
  \end{minipage}
\end{table}

\end{document}

I suggest you (a) keep loading the caption package and (b) employ a threeparttable environment to encase both the \caption directive and the tabular environment. It measures the width of the tabular environment and limits the width of the caption string to that width.

\documentclass{article}
\usepackage{caption}
\usepackage{threeparttable} % see https://ctan.org/pkg/threeparttable
\begin{document}
\begin{table}[h!]
    \centering
    \captionsetup{singlelinecheck = false, justification=raggedright}
    \begin{threeparttable}
    \caption{System specifications}
    \begin{tabular}{@{} ll @{}}
         Processor & Intel(R) Core(TM) i7-7500U CPU @2.70\,GHz 2.90\,GHz \\
         Installed RAM & 8.00 GB (7.89 GB usable)
    \end{tabular}
    \end{threeparttable}
\end{table}
\end{document}