I am a dual citizen - US and Russia, my permanent residency is in US and I pay US taxes. However, I have a flat in Russia which I am considering to sell. I am aware of Russian tax law, but not sure how profits from this transaction would be processed in US. Do I pay any taxes in the US (I owned this property for over 20 years)?

On your return, you must report your world wide income. You will report property sale on schedule D (Form 1040). If you paid taxes in the foreign country, you will claim credit for taxes paid in the foreign country by filing Form 1116
Read: Your U.S. Tax Return: U.S. Citizen or Resident with Foreign Income