You don't have to, if you really need to prune, you just have to make sure that the wallet is loaded before syncing Bitcoin Core so it won't have to use rescan.
In your case, you wont have to resync again after it's done since you've now loaded the wallet (see if it's selectable from the "Wallet:" drop-down menu at the right).

You can use pywallet that's suggested by NotATether.
Or natively, use dumpwallet command to export all of the keys plus their respective addresses but you have to start Bitcoin Core first with legacy as the default address type because the dumpwallet file's addresses will be based from it; because you've mentioned that the wallet is from 2011~2013.

If you want to do it:

• Open Bitcoin Core's settings: "Settings->Options...->" Click "Open configuration file"
• Type in the text editor that will open: #addresstype=legacy and save it, then restart Bitcoin Core.
• Open Bitcoin Core, go to Console: "Window->Console", select the correct wallet from the drop-down menu of the console.
• Type the command: walletpassphrase "your_password" 120 to unlock the wallet (do not if it's not encrypted | "120" is to unlock it for 2minutes)
• Within the specified time, type dumpwallet walletdump.txt to create a dump file, you can change the name of the file if you want or specify the absolute path if you want to save it to another directory. e.g.: dumpwallet E:\folder\walletdump.txt.
• Find the file in Bitcoin Core's installation directory or the specified absolute path and open it with a text editor.

The addresses (and their private keys) should be listed there.
Be careful with the file's contents since it contains the private keys which be used to spend your wallet's funds.