Well, right off the bat that sounds like a clear cut case of using a nuclear bomb to kill a fly, it'll work, but why do it?
The joystick you are talking about has the ability to operate 8 functions, 2 on each of 4 switches.
The actuator has 2 functions. Extend and retract.
All you need is just one 3-way switch to do the job. In electric terms it's called a "DPDT (double pole, double throw) center off momentary contact" switch. This is basically a 2 function 'joystick' switch. When you push it one way the actuator will extend, when you push it the other way it retracts and when you let go it stops.
It will have 6 terminals on the back, wire it up so that the 2 leads from the actuator connect to the center 2 terminals. Then wire the positive power supply to the lower left and upper right terminals, and the negative to the remaining (lower right & upper left) terminals.
Does your brain hurt yet?
Seriously, any problems give a shout, it's really not hard to do.