I did something kind of like this when I was in college. I signed up to be an intern for one of my professors over the summer and basically I shadowed her in her studio and attended a convention with her (my first time tabling at a con!) and received feedback on the comic I was working on. In exchange for her time and advice I did flat colors, scanned stuff, took dictation, etc. This was a great experience for me and I learned a whole lot, but I think part of the reason it worked was because we were both really clear on what we wanted out of the experience. I'm also not sure how it would have worked out if it had not been in person, as most of the lessons I learned were just from watching her work... doing things she didn't even consider lessons like updating her blog, taking phone calls, answering emails, and organizing her workflow.
The issue I could see with the program would be that the apprentices would expect too much of their mentors and possibly become frustrated? They would need to go into the experience with very clear expectations and motives. Like: I want to learn how to be more organized, I want to learn how to promote on social media, I want to get better at layouts, I want to write better scripts, so that their mentor would know what to focus on during the limited time they'd be interacting (a few hours per week?)
I also don't think it could be randomly assigned, as I'm sure there will be more people interested in being mentored than people with the time to mentor. I would suggest having the list of mentors up and having the people interested in becoming an apprentice write a letter to the mentor/mentors they'd be interested in learning from and then letting the mentor pick who they think they could help the most. People will learn most effectively from people they want to learn from... and the more experienced artists know their own strengths so they could choose who they could mentor most effectively.
I also do think there should be some incentive for the mentor. In traditional mentoring, the apprentice usually works for the mentor for a reduced rate or free (or even pay the mentor in some cases like tattoo artists and in salons, but that's a bit much). I don't think it would be out of line to have the apprentice do flat colors or something for the mentor since the mentor is going to be putting their time into the relationship.
EDIT: if this is tl/dr, just read what Sarrowsmith posted above, lol