My house closed on Jan 2, 2020. All my money was taken out in Dec 2019, but technically dated of purchase is 2020. How do I claim first time buyer for the 2019 tax year?

As your actual date of purchase is not until 2020, you would not be allowed to claim the Home Buyers Amount until next year on your 2020 tax return.

CRA states "You can claim $5,000 for the purchase of a qualifying home in the year".

Line 31270 – Home buyers' amount