There are dozens of ways to accomplish remote control depending on your environment. I think I’m familiar with most of the unreliable ones :D
They fall into four categories (although arguably these are just variations upon RF/Wired):
- Low-level RF like XBee
- Line-of-sight: infra-red
For the actual mechanism you should need only a power supply- be it a battery or a conveniently placed mains adaptor- a small micro-controller board, a transistor and a motor.
Because you want the rabbit to pop up slowly and drop back immediately, you only need the motor to turn in one direction. Using a snail drop cam:
You can run the motor in test and figure out how long it takes for a slow rise and drop, then program the microcontroller to run the motor for that much time.
Adafruit have a nice little tutorial about using a transistor to drive a motor from an Arduino: https://learn.adafruit.com/adafruit-arduino-lesson-13-dc-motors/transistors
If you’re not so worried about penny pinching (this is the cheapest solution I can think of) then you can use any of the vast number of motor driver boards- picking one that can be used for another project in future might be a good idea.