Private developer charges you task wise or on hourly bases.
But here as I read you just need to update existing theme so there is no need to pay $180 for it again you can get updated version of theme.
Just visit theme page and follow steps, a message will popup there that you already purchased this theme or similar.
After following all steps a updated version of theme will installed in your store.
You can check it's preview mode and make necessary changes before making it live.
Note: If there is any custom code in your existing version of theme it will not available in updated version. And if you need that custom code you have to manually copy that custom code to new version of theme.
And rest about 'Manage Private Apps' there is no concern of it with your live theme.
Thank you for your response, Guleria. Am I supposed to pay Shopify or the developer the $180? According to Shopify notes, I am supposed to create a private app so that he can start working on my upgrade. At this time, I have not paid the $180. I think the money should be paid to Shopify. Then I guess that the developer has to become a Shopify partner and sign an ageeement to work on the site. Is that correct?
You can hire a Shopify Expert and that option is available in your store dashboard right side at top.
But again there is no concern of private app to start working. Private app is different thing.