Using `sed` in Android without `g` modifier?

In order to limit the replacement to a portion of the file up to and including the first matching instance, you can prepend the substitute command with a range of the form 1,\#pattern# i.e.

busybox sed '1,\#</verse-sub-section># s#</verse-sub-section>#</section>#' file

Note the use of the backslash escape before the alternate delimiter \# except where it is introduced by the s command.

Note that this will not have the desired behaviour if a match occurs in the first line; GNU sed supports a range of the form 0,\#pattern# to handle this situation, but busybox sed - at least the version I am able to test, BusyBox v1.22.1 (Ubuntu 1:1.22.0-8ubuntu1) - does not appear to.

Your question is ambiguous.  If you want to find the very first instance of </verse-sub-section> in the entire file, and replace it with </section>, then see the other answer(s).  My interpretation is that, given the input

1    <section>Genesis</verse-sub-section>
2    <verse-sub-section>In the beginning, God ...</verse-sub-section>
3    <section>Psalm of David</verse-sub-section>
4    <verse-sub-section>David Weeps</verse-sub-section>

you want to replace </verse-sub-section> with </section> on lines 1 and 3.  If that's your question, the answer is

sed -i 's#\(<section>.*\)</verse-sub-section>#\1</section>#' file

Putting <section> in the old field of the s/old/new/ command guarantees that only lines containing <section> will be processed.  The \(…\) delimits a part of the line: the part starting with <section> and going up to (but not including) the </verse-sub-section>.  And the \1 in the new field causes the identified substring of the matched string to be inserted back into the file -- i.e., left untouched.

At least that's how it works in normal versions of *nix.  Android shouldn't be any different, but I can't guarantee that it will work in busybox.