Is it possible to show a pop-up or drawer cart after adding products to the cart with Also Bought?

In order to add products to the shopping cart, the app uses the Shopify Ajax API, which is indeed the official way to do it.

However, some themes also use pop-up or drawer carts and, given the large variety of themes in Shopify, there isn't yet any standard mechanism to handle these dynamic carts in every theme.

For that reason, we must support each pop-up or drawer cart individually in our code, what it is not always possible since the theme files may be inaccessible to us, minimised or obfuscated. However, for the most popular themes, the app will open the pop-up or drawer carts correctly. We will continue to improve the support for pop-up carts and other features in future versions of the app.

With regard to displaying the shopping cart contents, Also Bought offers two options.

On the one hand, you can redirect your customers to the cart page after adding a product from the recommendation widget just by checking the option Redirect to cart page on the app preferences page.

If the Redirect to cart page option is disabled, 

your customers will remain on the product page and the clicked button will display the text 'Added' which by the way, is configurable from the app preferences page.

Additionally, for the most popular themes in Shopify the shopping cart icon will be updated and the pop-up or drawer cart will be opened if it is configured that in the store theme settings, replicating the behaviour from the product page (add to cart button).