How can I make my 1:1 session with mentor more effective?

What do you need help with with regards to your promotion. Have you identify your gaps? Can the mentor share his experience in that? If it is a mentor that work closely with you, they may be able to provide you their observation and room of growth as well. And what specifically are you trying to learn from the mentor?

Feel free to also use the mentor as a sounding board to bounce idea off. They would generally provide you feedback based on their experience and how they may approach it differently.

When you feel like you went through a tough situation or tough decision, you could also run it by the mentor to understand how they would approach it.

Don't think of questions on the day itself, it is hard. Start taking notes of what you are doing daily, as a daily reflection kind of thing. That way you are tracking the highlight and low light of your day. You can use the lowlight as input and phrase it to questions for your mentor and learn how they would approach it.