How to make formula in cases environment and a tikz figure side by side?

Here's a simple way to fix it (more info in the commented code lines):

\documentclass{article}

%\usepackage{xeCJK}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{pgf, tikz}
\usepackage{pgfplots}
\usepackage{caption}
\usepackage{subcaption}

\begin{document}
    Hello, world!
    \begin{figure}[!h]
        \begin{subfigure}[c]{.5\linewidth}% use c(center) instead of b(bottom) for relative vertical alignement
            \centering
            \begin{equation*}
                \begin{cases}
                    f(\sqrt{-\frac{a}{3}}) &< 0\\
                    f(0) &> 0\\
                    f(1) &< 0
                \end{cases}
            \end{equation*}
            \vspace{5ex}%for fine tuning the vertical position of <cases>
        \end{subfigure}%no empty line between subfigures environments
        \begin{subfigure}[c]{.5\linewidth}% use c(center) instead of b(bottom) for relative vertical alignement
            \centering
            \begin{tikzpicture}[scale = 0.5]
            \begin{axis}
            \addplot {x};
            \end{axis}
            \end{tikzpicture}
            \subcaption*{This plot shows us that ...}
        \end{subfigure}
    \end{figure}
\end{document}    

You don't need figure at all costs: if this is to be in an equation* environment, use it.

The simplest way I see is using a two column tabular, with m type columns.

\documentclass{article}

\usepackage{array}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{pgf, tikz}
\usepackage{pgfplots}

\usepackage{lipsum}

\begin{document}

\lipsum*[2]
\begin{equation*}
\begin{tabular}{
  @{}
  m{.5\textwidth}
  @{}
  m{.5\textwidth}
  @{}
}
\centering
$\displaystyle
\begin{cases}
  f(\sqrt{-\frac{a}{3}}) &< 0\\
  f(0) &> 0\\
  f(1) &< 0
\end{cases}
$
&
\centering
  \begin{tikzpicture}[scale = 0.5]
  \begin{axis}
  \addplot {x};
  \end{axis}
  \end{tikzpicture}
\tabularnewline
&
This plot shows us that something happens
and we're very happy about it
\end{tabular}
\end{equation*}
\lipsum[3]

\end{document}

An alternative with using tabularx package:

\documentclass{article}

\usepackage{amsmath,amssymb}
\usepackage{pgfplots}
\usepackage{caption}
\usepackage{tabularx}
\renewcommand\tabularxcolumn[1]{m{#1}}

\usepackage{lipsum}

\begin{document}
    \lipsum[1]
\begin{figure}[ht]
    \begin{tabularx}{\textwidth}{X>{\centering\arraybackslash}X}
\[
    \begin{cases}
    f(\sqrt{-\frac{a}{3}})  & < 0   \\
    f(0)                    & > 0   \\
    f(1)                    & < 0
    \end{cases}
\]
    &  
        \begin{tikzpicture}[scale = 0.5]
        \begin{axis}
        \addplot {x};
        \end{axis}
        \end{tikzpicture}           \\
    &   \caption*{This plot shows us that ...}
    \end{tabularx}
\end{figure}
    \lipsum[2]
\end{document}

Above MWE gives: