I just tried the following and it worked.

Make sure your Admin -> Configuration -> My Store -> Display Cart After Adding Product = True

Open file:

/includes/init_includes/init_cart_handler.php

Look for the following:

PHP Code:
if (DISPLAY_CART == 'true') {
    $goto =  FILENAME_SHOPPING_CART; 
Change: FILENAME_SHOPPING_CART to FILENAME_CHECKOUT_SHIPPING

Once this is done and your customer clicks that button to add the item to the cart, it adds it and takes them directly to Checkout. Since Checkout requires a login process, it asked for a username/password on my site. Maybe yours is setup different and it will not ask.

Make a backup of that file before doing anything. If this doesn't work or it's wrong advice, you will be able to just restore the backup file and be back to where you are now.