You need to parse output.xml file generated by Robot Framework to get results of your test suite. Alternatively JUnit xml reporting file may be generated by using -x parameter:
robot -x junit.xml test1.robot
To update results in Zephyr test cases use ZAPI for Jira. Please check https://getzephyr.docs.apiary.io.
Execution status update
Find test case id (issue id) by test case name (issue key):
https://<jira_server>/rest/api/2/issue/<test_case_name>
Find execution id by test case id:
https://<jira_server>/rest/zapi/latest/execution?issueId=<test_case_id>
Request:
PUT https://<jira_server>/rest/zapi/latest/execution/<execution_id>/execute
Headers:
Content-Type: application/json
Body:
{
"status": "1"
}
Status "1" is for PASS.
Step status update
Find step id by execution id:
https://<jira_server>/rest/zapi/latest/stepResult?executionId=<execution_id>&expand=
Request:
PUT https://<jira_server>/rest/zapi/latest/stepResult/<step_id>
Headers and body the same as above.