There are two ways to update ScriptRunner for Jira:
Version History on the Atlassian Marketplace
To upgrade ScriptRunner for Jira, navigate to Version History. From there, you can select a version to Download.
You can also click Watch in the upper right-hand corner to be notified when new versions are released.
Manage Apps in Jira Instance
If you navigate to Administration > Manage Apps, you can update by selecting Update.