In the snippet mentioned in the question, regex search pattern is incorrect. there is no <caption> in the input. its <caption id....
Second using preg_replace doesn't serve any purpose here. preg_replace expects three arguments. first should be a regex pattern for search. second the string to replace with. and third is input string.
Following snippet using preg_match will work.
<?php
//The input string from API
$inputString = '<caption id="attachment_1342" align="alignleft" width="300" caption="Cheers... "Forward" diversifying innovation to secure first place. "></caption> A group of 35 students from';
//Search Regex
$pattern = '/<caption(.*?)<\/caption>(.*?)$/';
//preg_match searches inputString for a match to the regular expression given in pattern
//The matches are placed in the third argument.
preg_match($pattern, $inputString, $matches);
//First match is the whole string. second if the part before caption. third is part after caption.
echo $matches[2];
// var_dump($matches);
?>
if you still want to use preg_match_all for some reason. following snippet is modification of the one mentioned in question -
<?php
//Sample Object for test
$obj = array(
array(
'title' => 'test',
'content' => '<caption id="attachment_1342" align="alignleft" width="300" caption="Cheers... "Forward" diversifying innovation to secure first place. "></caption> A group of 35 students from'
)
);
echo "<table border='1'>";
echo "<td>".$obj[0]['title']."</td>";
echo "<td>".$obj[0]['content']."</td>";
echo "</table>";
$html = $obj[0]['content'];
//preg_match_all will put the caption tag in first match
preg_match_all('/<caption(.*?)<\/caption>/s', $html, $matches2);
//var_dump($matches2);
//use replace to remove the chunk from content
$obj[0]['content'] = str_replace($matches2[0], '', $obj[0]['content']);
//var_dump($obj);
?>