How brief I should be when contacting a professor for a PhD position for the first time? I happened to find a PhD project that would align with my past experience really well. I would like to contact the professor responsible for the project. How brief should I be in my first email?

In addition to introducing myself and referring to the position, I would like to ask at least two questions about the project, and mention that I have been working on a similar project.

Is that enough? How would I finish the email then in a proper way? Just something like 


  If you are not too busy, I hope you could discuss the matter further. Attached is my CV.


or something entirely else?

I have a very good publication record, have received big prizes, lot of experience, etc. in my pocket but I guess I should not repeat that in the email, since it is in the CV?
Professors receive a number of such emails. You must get the professor's interest in the opening few lines.

"I am a prize-winning A+ student in the department of X at the university of Y. My most recent work - my honours study of 
Z - is very relevant to the advertised PhD position. I am very keen to talk to you about this opportunity, by email, phone or Skype, whenever would be convenient with you".

And then by all means ask your questions, that is good, it shows engagement. You could also expand on why your previous work and/or skills will be valuable to the professor.

You could get another professor to proof read your email to see if it hits the right note.