As I want to apply for a Ph. D position after my final thesis work (as I really, really, really am interested in the project), should I try to work along them or should I voice my disapproval as mentor for my thesis work?
My opinion would be to ask for a different student mentor to that professor and try to be assertive when asking. You shouldn't say anything unpleasant about that student you can't stand. When the professor will ask you why, just say the truth: you have a bad feeling about this and prefer a different mentor in order to keep the good work and good atmosphere in the lab. If the professor says yes, good. If not, it doesn't cost you anything to try.
Yet, this is just my opinion about a situation I don't know in detail. So, if in doubt, get some advise from someone you trust and who knows the situation.
how important is the mentoring for my final project?
Quite important I think.
- A good mentoring can make you save time by learning things faster and bring some insight concerning your work. According to my personal experience when I was doing my PhD: I had no mentoring and I was working mostly on my own. My supervisor was often away and lab mates had different topics to work on. It is during the thesis writing, at the end of my PhD, that I truly came to understand my research subject in depth, that I saw the mistakes and missing parts of my work, that I saw potentially interesting paths I could have taken... With a good mentor who can point to you your own weaknesses, it can transform a decent thesis into a much better one.
- With a bad mentor, you have to wrestle against both you research experiments and your mentor (who could be giving you ridiculous advises, who could spread your work into countless ideas with the time to finish nothing at the end,.). So, you need twice more effort to achieve the same result than if you had a standard mentor.