That is not a motor, that is a linear actuator. It is designed to be all the way in or out from the spring, and all the way the opposite direction when energized. Is that what you need?
All that needs is a transistor to energize it. Logic Level N-channel MOSFET would be good, such as https://www.digikey.com/products/en?keywords=aoi514
RDS(ON) (at VGS = 4.5V) < 11.9mΩ
Here's the kind of info you're looking for in the part you select:
A solenoid is no use for a syringe - it will either be too weak and nothing happens, or it will empty the syringe
in one go completely uncontrolled.
You need a linear actuator. An actuator is a motor and lead-screw mechanism. It will not be very cheap, this
is precision miniature engineering. The only way to be cheap is make the mechanism yourself, which requires
machine-shop facilities really. Standard studding (aka all-thread) is fine for making leadscrews if you are happy
for a little backlash.
Lots of people have done this, go and search around for inspiration.