If you are fine with not graying out the image, you could simply add a \only<4->{\color{gray}} in your slide like this:
\documentclass{beamer}
\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{xcolor} % this is needed as well
\begin{document}
\begin{frame}{My Frame}
\onslide<1->{
{
\only<4->{
\color{gray}
% see https://tex.stackexchange.com/a/236520/128658 for different nesting levels
\setbeamercolor{itemize/enumerate body}{fg=gray}
% see https://tex.stackexchange.com/a/388900/128658 for different nesting levels
\setbeamercolor{itemize item}{fg=gray}
}
Intro line
\onslide<2->{
\begin{columns}
\begin{column}{.49\textwidth}
\begin{center}
Pic goes here
\end{center}
\end{column}
\begin{column}{.49\textwidth}
\begin{itemize}
\item Very important thing
\onslide<3->{
\item Even more important thing
}
\end{itemize}
\end{column}
\end{columns}
}
}
}
\onslide<4->{
% After this pause, the previous part should be grayed out.
Big sentence
\begin{itemize}
\item Cool solution \pause
\item Strong statement
\end{itemize}
}
\end{frame}
\end{document}
This way the color-directive (which only acts on the current scope (aka the brace-enclosed region) will only be present from the fourth slide-part onwards (thus after the third \pause).
EDIT: I suppose for graying out the image you could do a similar trick in order to tweak the alpha-value of the image on the fourth slide-part. For inspiration see e.g. Includegraphics: set image opacity
EDIT2: As it turns out pause does not seem to deal graciously with extra scopes on the slide (or I failed to use them properly). Thus I have replaced the \pause commands with explicit \onslide blocks. Furthermore beamer uses some special magic to figure out the font color of text in list environments. In order to change these, you'll have to change the respective color in the current beamer theme. In much the same way, the bullet points themselves can be colored as well.
And finally I replaced your minipage constructs with beamer's columns environment (which was made for the exact purpose you seem to be using minipages now).