If you want to use core as your wallet furthermore you will have to sync the blockchain.
But if you just want to export the private keys and import them into another wallet (e.g. electrum[1]) you can using these commands in the console of core:
walletpassphrase YOUR_PASSWORD
to unlock your wallet (if encrypted), and:
dumpwallet c:\path\to\your\dumped\privkeys.txt
to dump all of your private keys into a .txt file.
Afterwards you can copy all of your private keys from the .txt file and import (of better: sweep) them into electrum.
To do this you have to (after creating a new wallet in electrum) go to the Wallet menu -> Private keys -> sweep.
You can paste all your private keys at once (one per line).
After sweeping your coins, all of your btc's will be sent to the electrum wallet (No longer backed up by your core-backup).
So, in this case, make sure to backup your electrum-wallet (12-word seed phrase when creating wallet).
[1]
https://electrum.org/#home