Some/most airlines do have pilots who have a "part time" management function e.g. the Training manager or Technical Manager may well be a line pilot,
but that's a slow road to the top. IMHO if you take that career path you are probably not going to progress fast enough to get to the very top at all in anything but a small airline.
Given the ambition to climb the corporate ladder. I would of course, get my master’s degree during my pilot years.
If you are lucky your airline will be employing you to fly their aircraft, and make no mistake that will occupy a
lot of your time, especially if the proposed new European Flight Time Rules are adopted. I'm not saying it can't be done, I have no real idea of the workload involved in gaining an MBA or even if studying for the same is compatable with working full time as a line pilot
but I suspect you will need a very understanding employer if you are expecting to be able take significant amounts of time "off line" for something they will probably regard as a hobby. Given the current state of the airline industry I would concentrate on getting any academic study you want to do completed now, then concentrate on pilot training.
Which means I’ll be 27-28 when I’ve completed my training. Wouldn’t this be considered getting a bit ‘old’ to get your first FO job?
No.