How can I manually sync my prices or stock levels from Amazon to WooCommerce?

For seller fulfilled items you can do so the same way as you import products from Amazon: by processing an Merchant Inventory Report. 

Visit the  Import page and you will find the options to select whether you want to update prices and / or inventory for existing WooCommerce products when an inventory report is processed. Set these options to your preference and click the “Preview” button. You will see a summary of what is going to happen when you process this report – just follow the instructions to proceed.

For items fulfilled by Amazon it's different as these don't have a quantity in the Merchant Inventory Report - the column is just empty. To fetch the actual FBA quantity from Amazon, an FBA Inventory Report has to be requested and processed, which happens automatically once a day if you have enabled support for FBA on the general settings page.

You can manually request a fresh FBA inventory report on the Reports page at any time - which is highly recommended when you imported FBA items from Amazon and don't want to wait until the next FBA Inventory Report is requested automatically.