It depends on the air pressure inside your submarine and the maximum depth you want to operate at. Friction of the syringe will also have to be accounted for.
Lets say you want to dive 100meters max and the air pressure inside the unit is atmospheric (which cancels out the air pressure above the water).
We then calculate the hydrostatic pressure:
1000kg/m^3 *10m * 9.81m/s^2 = 98kPa
The force is dependent on the diameter of syringe. Looks like the diameter of a 200ml syringe is 44.75mm.
Area = 3.14159* ( 44.75mm/2 )^2 = 0.00157 m^2
Force = 98kPa * 0.00157 m^2 = 154N (~35 pounds force)
Note that this method considers the volume of the air in the submarine large compared to the volume in the syringe. The required force will be reduced by the increased air pressure in the submarine.
The static friction from the seal will also need to be measured empirically and added to the total force required. To estimate it, this link gives an overview of forces required for a range of syringes; recently-moved up to 3 year old. Based on their graph it looks like about 5 Newtons for a recently-moved (6 or 8mm diamete?) syringe. A 200ml syringe will be more than this and will need tested empirically with a scale.
Add the friction value to the total force required and multiply by a 1.5 safety factor.
Then just buy a linear actuator with the right force and stroke. Then just use the manufacturer specifications on power consumption as it already includes the gear/motor efficiency calculations. This FA-35-12-(X)-P linear actuator has a maximum draw of 5A at 12v or 60watts.