How to rename \pi as another value only for y axis without affecting \pi used in x axis?

I don't think there is (that is, I didn't find) a built-in way to do that. With etoolbox you can patch the code for the x and y labels to add arbitrary definitions:

\usepackage{etoolbox}
\makeatletter
\patchcmd\pst@@vlabels
  {\vbox to -\psk@ylabelOffset}
  {\YAxis@labelHook@code\vbox to -\psk@ylabelOffset}
  {}{\FailedToPatch}
\define@key[psset]{pst-plot}{ExecuteBeforeYLabels}[]{\def\YAxis@labelHook@code{#1}}
\patchcmd\pst@@hlabels
  {\ifcase\psk@xlabelPos}
  {\XAxis@labelHook@code\ifcase\psk@xlabelPos}
  {}{\FailedToPatch}
\define@key[psset]{pst-plot}{ExecuteBeforeXLabels}[]{\def\XAxis@labelHook@code{#1}}
\psset{
  ExecuteBeforeYLabels={},
  ExecuteBeforeXLabels={},
}
\makeatother

then you just need:

\psset{ExecuteBeforeYLabels={\def\pi{\sqrt{3}}}}

Full code:

\documentclass[pstricks,border={3pt 3pt 15pt 15pt},12pt]{standalone}
\usepackage{pst-plot,pst-calculate}
\usepackage[nomessages]{fp}

\FPset\EPSILON{0.3}

\FPeval\Const{root(2,3)}

\FPeval\Left{0-pi}
\FPeval\Right{2*pi}

\FPeval\Top{Const}
\FPeval\Bottom{0-Const}

\FPset\xTrigLabelBase{4}
\FPset\yTrigLabelBase{3}

\FPset\Dx{1}
\FPset\Dy{1}

\FPeval\dx{pi/xTrigLabelBase*Dx}
\FPeval\dy{Const/yTrigLabelBase*Dy}

\FPeval\XUnit{15/(Right-Left)}
\FPeval\YUnit{6/(Top-Bottom)}

\usepackage{etoolbox}
\makeatletter
\patchcmd\pst@@vlabels
  {\vbox to -\psk@ylabelOffset}
  {\YAxis@labelHook@code\vbox to -\psk@ylabelOffset}
  {}{\FailedToPatch}
\define@key[psset]{pst-plot}{ExecuteBeforeYLabels}[]{\def\YAxis@labelHook@code{#1}}
\patchcmd\pst@@hlabels
  {\ifcase\psk@xlabelPos}
  {\XAxis@labelHook@code\ifcase\psk@xlabelPos}
  {}{\FailedToPatch}
\define@key[psset]{pst-plot}{ExecuteBeforeXLabels}[]{\def\XAxis@labelHook@code{#1}}
\psset{
  ExecuteBeforeYLabels={},
  ExecuteBeforeXLabels={},
}
\makeatother

\psset
{
    xunit=\XUnit,
    yunit=\YUnit,
    dx=\dx,
    Dx=\Dx,
    dy=\dy,
    Dy=\Dy,
    xtrigLabels,
    ytrigLabels,
    xtrigLabelBase=\xTrigLabelBase,
    ytrigLabelBase=\yTrigLabelBase,
    showorigin=false,
    ExecuteBeforeYLabels={\def\pi{\sqrt{3}}},
    % ExecuteBeforeXLabels={\def\pi{\sqrt{3}}},
}

\begin{document}
\begin{pspicture}(\pscalculate{\Left-\EPSILON},\pscalculate{\Bottom-\EPSILON})(\pscalculate{\Right+\EPSILON},\pscalculate{\Top+\EPSILON})
    \psplot[algebraic,linecolor=red,plotpoints=100,linewidth=2\pslinewidth]{\Left}{\Right}{\Const*sin(x)}
    %\def\pi{\sqrt{3}}
    \psaxes{<->}(0,0)(\pscalculate{\Left-\EPSILON},\pscalculate{\Bottom-\EPSILON})(\pscalculate{\Right+\EPSILON},\pscalculate{\Top+\EPSILON})[$x$,0][$y$,90]
\end{pspicture}
\end{document}

\makeatletter
\let\pst@@VLabels\pst@@vlabels
\def\pst@@vlabels{\def\pi{\sqrt3}\pst@@VLabels}
\makeatother
\begin{document}
[...]
\end{document}