I am remodeling the kitchen and don't have an easy way to pull a new circuit from the breaker box. I do have a new 8/3 wire from the breaker box to the kitchen though. Can I run this to a metal box and then run a 12/3 (bx) through the inside of the cabinet for the new electric cook top?

You can, but it then needs to be on a 20 Amp breaker.

What is the rating of the cooktop??

What is the rating of the cooktop???
Same question I have??

Also, If you decide to do this, you may want to leave some sort of information letting someone know that the circuit is no longer a 40A circuit.

Being on a 20 amp 2 pole should clear up any misunderstanding as well as identify the panel. I would think 30 or 40 Amp for Cooktop. They do make 20 amp.
I think people would be surprised you could have a portable spa with heater, blower and jets, and work on a 15 Amp recepticle/250 Gallons?
I would like to believe Poster knew that #12 is what he needed. If he's not sure The Nameplate Rating will be helpful.
Only this Cooktop can be connected, and not any previous appliances.
The way I was reading this was for something, now it is for something else, the 20 Breaker would solve this, and the metal box needs to be grounded. Good Luck

Generally speaking, a range top will be 30 amps. A range/oven combination will be 40 amps.

With either type unit, the circuit should be an independent circuit and as such it should not be used as a feeder to at 20 amp, #12AWG.