Make stealth horizontal in TikZ?

The simplest option is to define an auxiliary coordinate:

\coordinate (start) at ($(a.north east)!0.2!(a.south east)$);

for later use:

\draw[-stealth] (start) -- (start-|b.west);

The complete example:

\documentclass[a4paper]{article}
\usepackage{tikz}
\usetikzlibrary{backgrounds, calc, positioning, fit}

\begin{document}

\begin{tikzpicture}
    \node[draw] (a) {A node};
    \node[draw,align=center, anchor=south west,xshift=0.5cm] (b) at (a.south east) {Another\\node};
        \coordinate (start) at ($(a.north east)!0.2!(a.south east)$);
    \draw[-stealth] (start) -- (start-|b.west);
\end{tikzpicture}

\end{document}

The result:

Note

I took the liberty to vertically align the blocks.

One solution is to create horizontal - (and a vertical |) to path style.

\documentclass[tikz,border=7pt]{standalone}
\usetikzlibrary{calc}
\tikzset{
  -/.style ={to path={(\tikztostart)--(\tikztostart-|\tikztotarget)\tikztonodes}},
  |/.style ={to path={(\tikztostart)--(\tikztostart|-\tikztotarget)\tikztonodes}}
}

\begin{document}
  \begin{tikzpicture}
    \node[draw] (a) {A node};
    \node[draw,align=center] (b) at (2,0.25) {Another\\node};
    \draw[-stealth]
      ($(a.north east)!0.2!(a.south east)$) to[-] (b.west)
      (a) edge[|] (b.north);
  \end{tikzpicture}
\end{document}