Send this person a brief e-mail, explaining the followings:
- How you found out about him/her, and give a 2-sentence introduction of yourself.
- That his/her career path and/or research interests overlaps with or inspires yours very much.
- Ask if you can have a phone conversation for 15 minutes to answer some questions. You can also ask for a meeting if he/she is nearby.
- List the brief questions that you plan to cover. Don't be too broad. Think if these 15 minutes are really the only time that he/she will be willing to talk to you, what would you like to get out from this?
- Provide 4-5 dates/times for him/her to pick, and invite them to suggest some dates if none of them works.
Avoid asking for reference letter, inside contacts, or any kind of favor beyond just formal career advices. He/she may not feel vested enough to do that, and if you push, he/she may close up. Also, don't attach any CV/resume; that would look like you're looking for a position.
And also don't ask "How did you get a job at an institution like this one?" Zoom out and ask for the job search process he/she went through. Ask him/her to elaborate on the thought process and how the pros and cons were weighed. Once the topic gets going, you can probe a bit further, but the focus should be on the faculty member, not the institution.
On the date, be on time, and honor the 15 minutes (or whatever you both agree upon) limit. At the end of the meeting, ask if you may ask for his/her opinion very occasionally through e-mail in future, and then establish the mentoring relationship from there.
When you got home, follow up with a thank you e-mail.
The tricks are:
- Don't ask them to write back. Replying e-mail on this kind of issues takes a lot of time and thought, plus he/she may not know you well enough to know if the recommendations are suitable for you.
- Make it low stake. At most they'll lose are 15 minutes.
- Make it thoughtful by listing highly relevant questions. This shows that you really did look at their work and know something about it.