Hi,
It's possible yes.
To not display out of stock products, you can simply set the "display out of stock products" option of the configuration.
And if you want to restrict the add to cart button to only a product, you can restrict the price of the product to a specific user group which will be added only to that user. Then, make sure that the "display add to cart button for free products" option is turned off in the configuration so that other users don't see the add to cart button.
That means however that they won't see the price of the products either.
So one solution you could do would be to hide the normal price display and instead add the price in the description of the products or a custom product field.
Note that price restriction per user group is only available with the Business edition.