How do I align summation signs instead of their subscripts?

I would avoid so long subscripts, breaking them into two lines.

You can get the alignment with the help of eqparbox; the first argument to \mathbox is a unique identifier, which must be different for every usage.

I'd also use alignat, in order to avoid excessive space between the equation and the related condition.

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{eqparbox}

\newcommand{\mathbox}[3][\mathop]{%
  #1{\eqmakebox[#2]{$\displaystyle#3$}}%
}

\begin{document}

\begin{alignat}{2}
\mathbox{A}{\sum_{\substack{j\in\mathcal{V}' \\ (i,j)\in\mathcal{A}}}}x_{ij}^E
  &\le 1,
  &\qquad& \forall i \in \mathcal{R}',\\
\mathbox{A}{\sum_{\substack{i\in\mathcal{N}_E\setminus\mathcal{N}_{342} \\ (i,j)\in\mathcal{A}}}} x_{ij}^E
  &= 0
  &\qquad& j \in \mathcal{R}_s',
\end{alignat}

\end{document}

You can use \smashopeorator[l]{...} from mathtools for this. Here I've aligned on the sums, the \le,= and the consitions, using alignat for better spacing.

Note also that I use \setminus not \backslash. \setminus has proper status (don't remember if it is a binary operation), whereas \backslash is a normal symbol (aka no automatic spacing).

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,mathtools}

\begin{document}

\begin{alignat}{3}
  &\smashoperator[l]{\sum_{j \in \mathcal{V}': (i,j) \in \mathcal{A}}}
  x_{ij}^E &\le 1, &\qquad& \forall i \in \mathcal{R}',\\
 &\smashoperator[l]{\sum_{i \in \mathcal{N}_E \setminus
     \mathcal{N}_{342}: (i,j) \in \mathcal{A}}}
 x_{ij}^E &= 0 && j \in \mathcal{R}_s',
\end{alignat}

\end{document}