In the catalog search, only name & price is showing up in the results, but I want to have the "add to cart" button also.
Is it possible and if yes then how to do it?
By default products in the catalog search will have the "Add to Cart" button. You may not able to see it for the following reasons:
Maybe the product is out of stock. Make sure the product is in stock and check whether you are able to add it to the cart
May be someone removed/commented the "add to cart" button in the catalog search template. You can check the file and uncomment it if it is commented out.
I would suggest few extension, which are much better for your store.
One I could remember is from Amasty.
https://amasty.com/search-pro.html
However, this doesn't has "Add to Cart" button but obviously you can add it later.
It's not a complete answer as you wanted but should give you an option.