Having arrows enter and leave from different parts of a node in tikzcd

You can define start style that use to path to set the starting anchor.

\documentclass[tikz,border=7pt]{standalone}
\usepackage{mathtools}
\usetikzlibrary{cd}
\tikzstyle{start}=[to path={(\tikztostart.#1) -- (\tikztotarget)}]
\begin{document}
  \begin{tikzcd}[]
    \text{Long text 1} \arrow[rr] \arrow[d,start=205]
                  &               & \text{Long text 2} \arrow[d,start=335] \\
    {A} \arrow[r] & {B} \arrow[r] & C
  \end{tikzcd}
\end{document}

Welcome to TeX-SE! You can do that using this answer. I'm not sure, though, I personally would go for it.

\documentclass{article}
\usepackage{tikz-cd}
\usepackage{amsmath}
\begin{document}  
\[  \begin{tikzcd}[execute at end picture={
    \draw[->] ([xshift=-4em]\tikzcdmatrixname-1-1.south) -- (\tikzcdmatrixname-2-1);
    \draw[->] ([xshift=4em]\tikzcdmatrixname-1-3.south) -- (\tikzcdmatrixname-2-3);
      }]
\makebox[1em][r]{Long text 1} \arrow[rr]  &    & 
\makebox[1em][l]{Long text 2} \\
{A} \arrow[r] & {B} \arrow[r] & C
  \end{tikzcd}\]
\end{document}