We can use listdata.svc to update list item based on the casenumber field. The following example code for your reference, modify the variables and add the code into a content editor web part, click the button to update list item.
<script src="https://code.jquery.com/jquery-1.12.4.min.js" type="text/javascript"></script>
<script type="text/javascript">
function updateListItem() {
var siteUrl="http://sp2010";
var listName="StatusList";
var caseNumber="456798";
var itemProperties={
'Status': 'Approval'
};
$.ajax({
url: siteUrl + "/_vti_bin/listdata.svc/"+listName+"?$filter=CaseNumber eq '"+caseNumber+"'",
type: "GET",
headers: {
"Accept": "application/json;odata=verbose",
},
success: function (data) {
var item=data.d.results[0];
$.ajax({
type: 'POST',
url: item.__metadata.uri,
contentType: 'application/json',
processData: false,
headers: {
"Accept": "application/json;odata=verbose",
"X-HTTP-Method": "MERGE",
"If-Match": item.__metadata.etag
},
data: Sys.Serialization.JavaScriptSerializer.serialize(itemProperties),
success: function (data) {
alert("succeeded.");
},
error: function (data) {
alert(JSON.stringify(data));
}
});
},
error: function (err) {
console.log(JSON.stringify(err));
}
});
}
</script>
<input type="button" value="update list item" onclick="updateListItem()"/>
