John, I am not familiar with "project manager" jobs at MBB" or "1st round interview with a Partner". The lower level of manager is a "Project Leader" at BCG, but your profile doesn't suggest to me that you'd be qualified for this quite yet => is there a significant part of your background that you are leaving out? I mean, kuddos to you if this is truly what you are in the running for, but it defies pretty much anything I've seen so far.
Generally speaking, all interviews have a mix of a case interview (for which you will have to prepare, they are much harder than what you've faced to enter Big4) and fit. Partner-led interviews might be a bit less structured (they own the company and basically do what they want), but even they typically still give a case. If you are truly interviewing for a PL position and meet a Partner in your first round though, all bets are off in terms of what to expect. I'd still highly suggest you do a solid case interview prep and work with a coach in particular. It's taken most of us well over 100 hours of dedicated prep to really understand how to crack a case, please don't dismiss this part.
In terms of the expectations, I'll again have to assume you'd be a PL at BCG. If so, the expectations would be higher at MBB: you will need to drive the project (small projects don't even have a Principal above you, so you become the face of BCG on a day-to-day basis); you will have to craft the story with the help of a partner, and mentor your consultants and associates + ensure they are all working towards a consistent message. Compared to a Big4, you are much more likely to have to create a new solution vs. reusing a cookie-cutter one, but will also benefit from solid mentoring inside the company as needed. Just don't hesitate to ask for help, especially at first.
Regarding your 3rd question lastly- yes, definitely yes: reach out to anyone you know there to get insights