score:3
You should bring both passports. But which one you use may be different with the airlines and with immigration control.
For instance, if you need to enter the Schengen area you need to give the airline your new biometric passport. When you return then you may need to give the airline both passports to prove that you have residence.
If you need to show Russian immigration that you have a residence permit, then you need to give Russian immigration your old passport with your residence permit. They might also need to see the new passport. This situation is common enough that they deal with it every day, and it should be no trouble to you.
Upvote:2
Which passport should I use to book airline tickets and which one should I present to Russian border guard in the airport?
Show your non-biometric passport to Russian immigration. If asked about Schengen visas, show them your biometric passport in addition to the non-biometric one.
As for the airline, it makes absolutely no difference what details you put in during check-in. As long as you have your biometric passport with you at the time of boarding, they'll let you on board.
Is it a strict rule that I should use the same passport when booking tickets and passing the passport control?
No, there is no such rule.
The only passport I can use to enter Greece is my biometric passport. The Greeks, however, may have questions because its completely blank.
Having a blank passport is not suspicious at all. At worst, you can show them your other Ukrainian passport if asked about it.