How can I use variables in the LHS and RHS of a sed substitution?

You could do:

sed "s/$old_run/$new_run/" < infile > outfile

But beware that $old_run would be taken as a regular expression and so any special characters that the variable contains, such as / or . would have to be escaped. Similarly, in $new_run, the & and \ characters would need to be treated specially and you would have to escape the / and newline characters in it.

If the content of $old_run or $new_run is not under your control, then it's critical to perform that escaping, or otherwise that code amounts to a code injection vulnerability.

This worked:

cat update_via_sed.sh | sed 's/'"$old_run"'/'"$new_run"'/'

As I want to 'reuse' the same file I actually use this for anyone wishing a similar approach:

cat update_via_sed.sh | sed 's/'"$old_run"'/'"$new_run"'/' > new_update; mv -f new_update update_via_sed.sh

The above created a new file then deletes the current file than rename the new file to be the current file.

You can use variables in sed as long as it's in a double quote (not a single quote):

sed "s/$var/r_str/g" file_name >new_file

If you have a forward slash (/) in the variable then use different separator like below

sed "s|$var|r_str|g" file_name >new_file