Well, since I have physics exam in a cople of hours with problesm such as this one, I decided to make a more or less detailed solution.
First let's take a look at the diagram. On it we have 3 major points: weight, pulley wheel and the shutter.
I'll start with the wheel and assume that it will rotate in the negative mathematical direction. Wheel's radius is R and its mass is Mw.
First we should take a look at the forces acting on it. We have the weight of the wheel Qw=Mw*g actung downward and we have the rection of the surfece, N acting upward. They cancel eachother out. Next, we have rope tension forces. S1 is the force which tenses the rope from the weight side and S2 is from the shutter side.
Now we need to write the moment equation for the system.
Wheel's moment of inertia is Iw=Mw*R^2/2 and its angular acceleration is α.
For the system to work, we need the α to be greather than zero. So we get the obivous α=(S1*R-S2*R)/Iw from which we can deduct that S1 should be greater than S2. We also know that α=a/R. We'll need that later. Next interesting thing which we can find from the wheel is the ratio between two forces. S1=S2+Iw*α/R. Also, the force S2=S1-Iwα/R
Next, let's take a look at the shutter. Let's say that the length of the rod is L2. We have force S2 which is pulling the rod upwards and it's creating the moment of force (torque) which gives us: M2=L2*S2. We also know that we have a minimum moment of force needed to trigger the mechanism and we'll calculate that later.
Lastly we have the weight side. We want to replace the weight with a motor and we need to calculate its moment of force. The motor has axle whose radius is r. We know that mass of the weight is m=12.8g. We can see that the force S1 which tenses the rope in the wanted direction is S1=Q1=m*g=0.125568 N or 125.568 mN. To achieve such a force, we'll need moment of force which is equal to M1=Q1*r, where r is the radius of the motor's axle.
Now let's take a look at what we can tweak. It would be good is we could get the needed moment to operate the shutter. If we know it, we can twaek the rest of the system.
So by taking a look at the wheel equation, Iw*α=S1*R-S2*R, we see that the wheel should be as light as possible and that that its diameter should be as small as possible in order to minimise the losses.
After that, if we take a look at the moment equation for the shutter, we have M2=L2*S2. We can increase the length of the rod in order to minimize the needed force.
In the end, let's take a look at the weight side. We know that Q1=125.568 mN and that M1=Q1*r. So if we divide the torque of the motor from the list by the axle diameter, we can get the force it will produce and check if it will be good enough for our needs. Q1=M1/r If we take for example, ADM_1220, we see that it has torque of 4.1 mNm and axle diameter of 12mm. So we get F=4.1mNm/12*10^(-3)m=341.67 mN, which is enough to close the shutter.