You can do something like this:
\makeatletter
\newcommand*\labelcounter[2]{\begingroup
\protected@edef\@currentlabel{\csname p@#1\endcsname\csname the#1\endcsname}%
\label{#2}\endgroup}
\newcommand*\refsetcounter[2]{\setcounter{#1}{#2}%
\protected@edef\@currentlabel{\csname p@#1\endcsname\csname the#1\endcsname}%
}
\makeatother
The line with \protected@edef is taken from the definition of \refstepcounter. Usage:
\labelcounter{equation}{eq:123}
Full example:
\documentclass{article}
\makeatletter
\newcommand*\labelcounter[2]{\begingroup
\protected@edef\@currentlabel{\csname p@#1\endcsname\csname the#1\endcsname}%
\label{#2}\endgroup}
\newcommand*\refsetcounter[2]{\setcounter{#1}{#2}%
\protected@edef\@currentlabel{\csname p@#1\endcsname\csname the#1\endcsname}%
}
\makeatother
\begin{document}
\subsection{Bla}
\begin{equation}a+b\end{equation}
\subsection{Blabla}
\labelcounter{equation}{eq-test}
\label{sect-test}
Should be 0.2: \ref{sect-test}
\\
Should be 1: \ref{eq-test}
\end{document}
In your example, the variant with \refsetcounter would be more useful:
\documentclass{article}
\makeatletter
\newcommand*\labelcounter[2]{\begingroup
\protected@edef\@currentlabel{\csname p@#1\endcsname\csname the#1\endcsname}%
\label{#2}\endgroup}
\newcommand*\refsetcounter[2]{\setcounter{#1}{#2}%
\protected@edef\@currentlabel{\csname p@#1\endcsname\csname the#1\endcsname}%
}
\makeatother
\begin{document}
\newcounter{counterA} \setcounter{counterA}{0}
\refstepcounter{counterA}
\label{First_Label}
I get: \ref{Second_Label}. I would like 99.
\bigskip
\refsetcounter{counterA}{99}
\label{Second_Label}
I get: \ref{First_Label}. I would like 1.
\end{document}