You can give them a check. That's the simplest way to handle it.
On the W-2, boxes 3 and 5 are the amount of wages subject to tax. The actual taxes are in boxes 4 and 6. Box 4 should be 6.2% of box 3, and box 6 should be 1.45% of box 5. Box 3 should not be more than $142,800 (for 2021), so box 4 should not be more than $8,853.60. Box 6 could be more than 1.45% if box 5 is more than $200,000. The amount that you repay should be equal to box 4 plus box 6.
You can pay back the employer anytime that you and they agree on. You should do it as soon as you can. If the FICA taxes had been correctly deducted from your pay, you would have paid it already. And the employer is not going to wait until the end of the year to send the payment to the government. The only thing you should be cautious about, since they seem to be unfamiliar with the procedure, is to make sure, before you pay them, that you and they have a clear understanding about what is going to be on your W-2, and that they are going to pay the FICA taxes to the government on your behalf.
You don't pay the FICA taxes when you file your tax return. You are paying the taxes now by paying the money back to the employer. They will send it in to the government, the same as they would if they had withheld it from your pay.
(Oddly, Revenue Ruling 2009-39 doesn't say anything about the employer getting the money back from the employee. It just talks about the employer paying the FICA taxes. But it makes sense that they would ask you to pay back the amount that should have been withheld from your pay.)
