MorganS:
Yes, the A4988 is good for all bipolar steppers less than 1.5A per phase.
No problem at all. Why would there be?
That looks like a neat motor that will do what you want.
Thank you very much Morgan. In the second question, I was worried about the increase in the Arduino processor's temperature, which might happen due to plunger load.
Two hours ago, I asked someone who has made a syringe pump a long time ago and he said A4988 would not be a good option for my stepper motor (I didn't ask the exact reason); Instead, he used TB6600 as a motor driver beside the Arduino Mega 2560 R3. But A4988 has a lower cost and I can provide both of them to check whether it can be a nice choice for my project.
Robin2:
I don't follow the calculation.If the diameter is 4.69mm then the piston area is 4.69 * 4.69 / 4 * Pi = 17.276mm2
And if the stepper were to advance the piston by 0.00125mm then the change in volume would be 17.276 * 0.00125 = 0.0216 mm3. I think that is about 21 µL and if so it would require about 5 steps to dispense 100µL
(By the way I would not be surprised if I have got my maths wrong
)
Thank you again Robin. I made a mistake in calculating syringe area which was considering diameter instead of the radius! 
So I calculated again:
-if we suppose each "microstep+delay interval" would happen in 1 second.
-1 mm3/h=1 µl/h
EDITED: mm2 to mm3
FR=0.00125 mm/s * 17.276 mm2 * 3600 s/h = 77.742 µl/h
Robin2:
That stepper motor has an 18:1 gear reduction - which may be very beneficial for your project but you will need to consider carefully whether backlash in the gearbox would cause a problem with precision.In my calculations I had been assuming a stepper motor without a gearbox.
You are absolutely right. It might have a bad reaction at low speeds, but I am searching for that to see if I can use it. This way I think I can use the motor driver in full step.
Robin2:
PS ... to get repeatable movements of 0.00125mm you will need a VERY high quality mechanical support system.
Of course, it will be a really challenging section. But I am even pleased with lower precision than what we were talking about.
outsider:
The op said the screw pitch was 2 mm so I got 34.5513930838 ul per turn or 2.89423930773 turns per 100 ul per hour or 0.0482373217955 RPM or 2.89423930773 steps per second (3600 steps / rev) or 345514 microSeconds per step. :o Sound close?
Thank you "outsider" for your response.
I will use M6 screw with 1 mm pitch instead.
As Robin said it needs very high accuracy in mechanical assembly and it would be more important with that 3600 steps/rev motor. In fact, I know that I will not acquire these numbers but I can guess my perfect flow rate.