Can I Exceed a N Channel MOSFET's -Vgs Rating When Using It as a High Side Switch
Short answer: NO, you can never exceed the Vgs rating under any circumstances. To do so means you will break the oxide layer, and the MOSFET will no longer be a MOSFET - it will be garbage.
Driving the gate of a high-side N-channel MOSFET is not a new problem, and there exist many solutions. A couple of simple changes to your proposed circuit could be workable, refer to the red mark-ups in the image below:
What we need to do is prevent the gate voltage going high faster than the source voltage. A simple RC delay circuit may be sufficient for this, I have added R3 and C1 to your original circuit. The trick here is to make the time-constant at the gate drive to be much longer (slower) than any time-constant on the load side of the MOSFET. The time-constant of the load side is formed by the load device, and any parasitic capacitance it may have (as indicated by the dashed lines to C2), including the parasitic capacitances of the MOSFET.
Putting some typical values in:
Let's say that C2 = 1nF.
The time-constant on output is R1.C2 = 100Ω x 1nF = 100ns.
Let's make the time-constant at the gate to be 10 times this, ie: 1us.
At turn-on, the time-constant is R3.C1, so C1 = 1us / R3 = 1us / 1kΩ = 1nF.
At turn-off, the time-constant ie R2.C1 = 10kΩ x 1nF = 10us.
Conclusion:
For both turn-in and turn-off, the time-constant of the gate-source drive is much longer (slower) than that at the MOSFET output terminals, (Source, and Drain), so we have achieved our goal. This means the gate-source voltage will not exceed the maximum rating at any time.
But there is a problem: R3 and R2 form a voltage divider, so the highest voltage that the gate can reach will be:
28V x (10kΩ / 11kΩ) = 25.45V, which is only 1.45V above the drain voltage of 24.0V.
Now, this may or may not be OK - it depends on what you're trying to do. If you want to switch much faster than this, then this is not the solution you need.
To explore other solutions, these links may be a good starting point:
Driving a P-channel MOSFET with high voltage input
https://electronics.stackexchange.com/a/710378/341959