The Arduino part seems extremely simple. The external hardware is what is going to take the effort to design and build.
You don't give any hints about the size of the device or the force, distance or speed requirements of these two actuators, but if you're able to use a plain old RC servo for either of them that would make the project a lot easier.
I'm always on the lookout for low tech solutions to problems like this. Guessing that you want the solenoid to be 'up' when the arm is moving one way, and 'down' when it is moving the other way, you could consider using a car windscreen wiper motor for the reciprocating movement, and limit switches to turn the solenoid on and off.