What's the best way of typing the following 58 equations into LaTeX?

Here's a possible implementation; the delimiters can be changed.

\documentclass{article}
\usepackage{amsmath,multicol}
\usepackage{xparse}

\ExplSyntaxOn

\NewDocumentCommand{\constraints}{omO{x}}
 {
  \IfValueTF{#1}{\begin{multicols}{#1}\centering}{\begin{center}}
  \egreg_costraints:nn { #2 } { #3 }
  \IfValueTF{#1}{\end{multicols}}{\end{center}}
 }

\seq_new:N \l__egreg_constraints_seq

\cs_new_protected:Nn \egreg_costraints:nn
 {
  \seq_set_split:Nnn \l__egreg_constraints_seq { \\ } { #1 }
  \seq_map_inline:Nn \l__egreg_constraints_seq
   {
    \__egreg_constraints_item:nn { ##1 } { #2 }
   }
 }

\cs_new_protected:Nn \__egreg_constraints_item:nn
 {
  \tl_if_blank:nF { #1 }
   {
    $#2\sb{\clist_item:nn { #1 } { 1 }} + #2\sb{\clist_item:nn { #1 } { 2 }}\ge1$ \\
   }
 }

\ExplSyntaxOff

\begin{document}

\constraints{
  1,2 \\ 3,4 \\ 5,6
}

\constraints{
  1,2 \\ 3,4 \\ 5,6
}[y]

\constraints[3]{
  1,2 \\ 3,4 \\ 5,6 \\
  1,2 \\ 3,4 \\ 5,6 \\
  1,2 \\ 3,4 \\ 5,6 \\
  1,2 \\ 3,4 \\ 5,6 \\
  1,2 \\ 3,4 \\ 5,6 \\
  1,2 \\ 3,4 \\ 5,6 \\
  1,2 \\ 3,4 \\ 5,6 \\
  1,2 \\ 3,4 \\ 5,6 \\
  1,2 \\ 3,4 \\ 5,6 \\
}

\end{document}

You can automate the job by exporting the Excel spreadsheet to a comma delimited file (for example subscripts.csv) and reading your data with the readarray package.

MWE

Assuming that the content of subscripts.csv is:

1,2
1,3
2,39
7,5
26,5

The following code:

\documentclass{article}

\usepackage{readarray}
\usepackage{forloop}

\newcommand\constraint[2]{$x_{#1} + x_{#2} \ge 1$}

\begin{document}

\readarraysepchar{,}
\readdef{subscripts.csv}{\MyMacro}
\readarray\MyMacro\MyData[-,\ncols]

\newcounter{MyCounter}
\forloop{MyCounter}{1}{\value{MyCounter} < \MyDataROWS}
{
  \constraint{\MyData[\theMyCounter,1]}{\MyData[\theMyCounter,2]}\par
}

\end{document}

renders:

Use Excel's CONCAT function.

It's probably easiest to do this in Excel, especially if it's a one-time thing. Excel has several text-manipulation functions, and I have often found it a very convenient way to generate several similar lines of text or code. CONCAT(text1,text2,...) concatenates a bunch of text and data values into one text output.

If columns A and B contain your i and j values, enter the following in cell C1:

=CONCAT("x_{", A1, "} + x{", B1, "} \ge 1.")

and hit Enter. You can then copy-paste the function all the way down column C and get the result you want. You can then select all the output cells in column C and paste into your text editor.