You can just export the private keys from Bitcoin Core, you don't need to sync the blockchain for that. Import them into Electrum, and see what you have.
Or did you buy/download this "wallet.dat"? Your story doesn't make much sense.
As suggested by LoyceV, you don't need to wait for bitcoin core to synchronize, after checking whether the addresses in this wallet.dat actually have funds (utxo), open the bitcoin console located at: Window > Console.
If this wallet.dat is really from 2020 (legacy-wallet), you can select the addresses with allocated funds by typing directly in the command line window that will open when you click on console, type "dumprizkey youraddress", e.g:
The text that returns (if successful) will be your private key, do this with the same command using another address instead and import it into the electrum wallet.
Another important detail that I forgot to mention is that if the wallet is encrypted, before specifying the command, you must enter the password within the console with the following command:
walletpassphrase "your-password" 60If you have many addresses with allocated funds, you can give the command: dumpwallet "URL of the path to be saved on the computer with defined name" e.g:
dumpwallet "C:\Users\yourusername\AppData\Roaming\Bitcoin\bitcoin-keys.txt"
This will produce a readable document with all addresses, private keys and metadata belonging to this wallet.dat. With it you can open the document with a text editor on your computer and search for the desired address directly in the search keywords (ctrl + f), with this you can import the private keys of the corresponding addresses in electrum which does not require downloading the entire blockchain as it connects with servers provided by electrum.
When choosing to import these private keys that contain the funds in the electrum software, during the wallet creation process, in electrum you must select that you want to "import private keys or addresses"
Another item to be checked is the type of address to be imported into electrum:
Address starting with 3: p2wpkh-p2sh
Addresses starting with 1: p2pkh
Addresses starting with bc1q: p2wpkh
When importing the private key into electrum and the funds are allocated to one of the address types in the example above, simply enter the address type and private key, example below:
p2wpkh-p2sh:L4uEv1fycMbuGdCc16VD2NYkXJvzkLP2CPZSuYTawewDXKAgrf2j
p2wpkh:L4uEv1fycMbuGdCc16VD2NYkXJvzkLP2CPZSuYTawewDXKAgrf2j
p2wpkh-p2sh:privkey (will import the address with 3 prefix)
p2wpkh:privkey (will import the address with the bc1q prefix).