To upgrade ScriptRunner for Bitbucket, navigate to Version History. From there, you can select a version to Download

The Download option, highlighted in the Version History page.

You can also click Watch in the upper right-hand corner to be notified when new versions are released. 

Manage Apps in Bitbucket Instance

If you navigate to General Configuration Manage Apps, you can update by selecting Update

The Update option for ScriptRunner highlighted in the Manage Apps screen.