How to update SOLR document?
If you're writing code:

search_api_track_item_change('entity_type', array($entity_id));


That will mark an entity as 'dirty', and it'll be (re-)indexed on the next run.

If you're not writing code, install the rules module, add a rule with an event of "A flag type has been flagged...", and an action of type "Index an entity", with the relevant parameters.