Capital improvements are not entered into the asset/deprecation section until the tax year that improvement is placed "in service". When you paid for it does not matter. When the work started or completed does not matter. But there is no way a capital improvement can be placed "in service" before it's completed.
The fact the work started in 2022 doesn't matter. The fact you may have paid for it in 2022 doesn't matter. If the work was not completed and placed in service until 2023, then nothing concerning the property improvement will be entered anywhere on your 2022 tax return. You'll enter it next year on your 2023 tax return, assuming the asset was actually placed "in service" in 2023.
