The simple answer is "no"
But there are consequences that must be considered to use the elevator safely.
As the elevator moves, the force of gravity changes. People on out-bound elevators (assuming they start feet-toward-the-rim, they'd quickly learn to do this) will be forced against the floor of the elevator as it ascends from the center. If it decelerates fast enough, people will break their legs. Note that part of the problem here is that despite wanting to be feet-toward-the-rim ASAP, the force of the elevator's acceleration will push them against the core-side of the elevator. They need to reorient as soon as the elevator stops accelerating and before it begins to decelerate.
People on in-bound elevators (and this one is the real trick) don't have an easy solution. They need to learn to rotate their bodies feet-toward-the-center as soon as possible because they're being moved along at the velocity of the elevator and almost no matter how slowly it stops at the core, they're going to hit the core-side of the elevator's interior. Unfortunately, they might have to reorient at pretty much the last second. If the elevator moves so slowly (or decelerates so slowly) as to guarantee no boo-boos, what's the point of having the elevator?
So, how could you do this?
I'm imagining padded, fitted, well-strapped-in, keep-your-arms-and-legs-inside-the-ride-at-all-times seating that is electronically controlled to reorient the passenger as the elevator moves. Nobody gets hurt, and nobody gets sued. If you're the kind of person to hurl on roller coasters, this might not be the best way for you to travel.
But, then again, if you were such a person, what the heck are you doing in space in the first place?
Edit: You changed your question. It didn't quite invalidate my answer, but I need to add a comment.
Yes, there's a problem with that configuration
People moving from the rim ("down" in the photo) to the core are going to be slammed against the ceiling because they failed to rotate during the trip. Unless you're proposing the inside of the cab is separate from the outside, and spherical rather than cylindrical, so that the whole inner cab rotates.
And just in case you missed it in comments...
@Robyn pointed out that there is the angular force of the rotation to contend with. The angular rotation is greater near the core than the rim. You can use this calculator to figure that out. It's non-trivial (that rotten movie The Martian lied to me! And I like that movie A LOT!). So, vertical forces, horizontal forces... space sure isn't easy on the stomach.