No, if that extra $1000 payment was applied to your 2019 balance due then it is not deductible on your 2020 tax return.  

The extra $1000 payment was $1000 more than what we owed for 2019. Specifically, we owed $8000 and ended up paying $9000. Can we get the extra $1000 back by deducting from what we owe for 2020?

You should be able to report this payment under Federal / Deductions & Credits / Estimates and other taxes paid / Income taxes paid / Withholding not already entered on a W-2 or 1099.  The withholding will be reported on line 25c of the 1040.

You may want to verify that the IRS has given you credit for this amount.  View or create an IRS Account.  See this.

Online Account is an online system that allows you to securely access your individual account information.  You may view: 

• The amount you owe, updated for the current calendar day
• Your balance details by year
• Your payment history and any scheduled or pending payments
• Key information from your most recent tax return
• Payment plan details, if you have one
• Digital copies of select notices from the IRS
• Your Economic Impact Payments (EIP 1 and EIP 2), if any