if only the top vertical space is mater of concern, than solution provided by Ulrike 's answer can help you:

\documentclass{article}
\usepackage{array}
\begin{document}
I don't want the extra vertical space preceding the bulleted list:
\begin{tabular}{>{\csname @minipagetrue\endcsname}p{4cm}p{4cm}} % insert minipages in cells
\hline % for seeing the top tables's border
\begin{itemize}
\item hello
\item hello also in the next item
\end{itemize}
& hello \\
\hline
\end{tabular}
\end{document}
However, if you like to control more attributes of list in table, than you can do this by use of enumitem package and define new list, for example tabitem:
\documentclass{article}
\usepackage{array}
\usepackage{enumitem}
\newlist{tabitem}{itemize}{1}% <-- defined new list for use in tables
\setlist[tabitem]{nosep, % <-- new list setup
leftmargin = * ,
label = $\bullet$ ,
after = \vspace{-\baselineskip}
}
\begin{document}
I don't want the extra vertical space preceding the bulleted list:
\begin{tabular}{>{\csname @minipagetrue\endcsname}p{4cm}p{4cm}} % insert minipages in cells
\hline % for seeing the top tables's border
\begin{tabitem}
\item hello
\item hello also in the next item where is no space after end of list
\end{tabitem}
& hello \\
\hline
\end{tabular}
\end{document}
It may be handy, if you define new column type, for example
\newcolumntype{P}[1]{>{\csname @minipagetrue\endcsname\RaggedRight}p{#1}}
and than write:
\documentclass{article}
\usepackage{array}
\usepackage{enumitem}
\newlist{tabitem}{itemize}{1}% <-- defined new list for use in tables
\setlist[tabitem]{nosep, % <-- new list setup
wide=0pt, % <-- new "makeup" for list formatting
% see @Bernard's comment below
label = $\bullet$ ,
after = \vspace{-\baselineskip}
}
\usepackage{ragged2e}
\newcolumntype{P}[1]{>{\csname @minipagetrue\endcsname\RaggedRight}p{#1}}
\begin{document}
I don't want the extra vertical space preceding the bulleted list:
\begin{tabular}{P{4cm}p{4cm}} % insert minipages in cells
\hline % for seeing the top tables's border
\begin{tabitem}
\item hello
\item hello also in the next item where is no space after end of list
\end{tabitem}
& hello \\
\hline
\end{tabular}
\end{document}
where is also considered @Bernard comment regarding list formatting. Above MWE (Minimal Working Example) gives:

Note: definition
\newcolumntype{P}[1]{>{\csname @minipagetrue\endcsname}p{#1}}
is equivalent to definition:
\makeatletter
\newcolumntype{P}[1]{>{\@minipagetrue\RaggedRight}p{#1}}
\makeatother
tabularvstabularx, I guess). Renamed my question to make the connection more explicit. – Roly Dec 12 '19 at 17:20tabularxbut definition of column type. It can bep{...}too! I will show you in my answer (asap)! – Zarko Dec 12 '19 at 18:41