Can I set a cron for deleting and re-indexing Search API daily?
If you have Drush on your server, you can use the drush search-api-reindex command. This way you don't need to invoke a cron, but can force a reindex.

Note: You'll need to run drush search-api-list first to get the numeric identifier for the index when you run the re-index command.

Reference more info here: http://drushcommands.com/drush-6x/search-api/search-api-reindex