If it's an early Classic, it'll have an ABR-1 bridge. But, there's a ton of Classics out there with the Nashville bridge. Best way to verify what you got is to look underneath the threaded thumbscrews and see where the posts go into the body. If the posts screw into a metal insert, the bridge is a Nashville. If they screw directly into the wood, the bridge is an ABR-1.
As for wrapping the strings around the tailpiece, you don't need a special tailpiece for that. Just lower the stock tailpiece all the way down to the guitar's top, and then load the strings from the front to the back. This is commonly known as top-wrapping.
Here's an example of Nashville bridge (notice the metal insert under the thumbscrew) with a lowered & top-wrapped tailpiece...