I appealed my 2017 property tax related to a rental. In 2018 I received a refund for the approved appeal. Do I need to file amended fed'l & state returns? If so, by when?

You are a cash basis taxpayer and you did not receive the refund till 2018.  It will be a refund of a prior year property tax. Use the amount to reduce the property taxes you can declare this year. It may not impact you at all if you only claim a standard deduction.

Yes, I am a cash basis taxpayer. To reduce the property taxes I declare this year, do I just subtract the refund from what I paid and enter that amount for property taxes? Or do I put the subtraction in a specific line item?  Or do I have to count the property tax refund as income?

Yes just reduce the amount paid by the amount you received as a refund.