Need a solenoid to pull a Nerf Gun trigger
Why a solenoid might not work
Solenoids in this situation would probably not be ideal. From my experience solenoids have great holding force, but not that strong initial force, so it may not be able to supply the needed 1Kg of force to get the trigger pushed. I have used beefy solenoids that run on high voltage and they have shown to be weak right at the start of motion to the point where you can hold the shaft in without much effort at all
Alternative control methods
A better alternative as some have already mentioned are rc servo motors, which depending on which one you get could definitely supply the needed force, and are easy to program with an arduino, as well as they run on low voltages depending on the one you get. Another alternative is a linear actuator which are available in many different forms. The simplest way would be attaching a motor to a screw (keeping positioning in mind) such as a stepper motor. Or there are already made linear actuators that have their own internal limit switches and all they need are a reversible DC current to control them. (using something like an H-bridge circuit). Despite those two as sticking out as the most obvious to me, there are definitely more ways to go about it.
Diy solenoid
Despite the issues with solenoids as stated previously, creating one from scratch is a hassle. Finding/making a good core, winding it up, only to realize that you too much wire, or that the resistance is too high for your application. I'm not sure as to the open-endedness of your project but from experience sticking commercial for something like a solenoid is usually the way to go.
Alternative Triggering Methods
Besides finding more ways to pull the trigger, replacing the trigger mechanism itself might be an option. Look on youtube and you will see plenty of videos of teardowns of your nerf gun. Once you see what is inside and making the trigger mechanism work it may occur that it is easier to just modify the trigger mechanism itself.
I'm not sure on size, voltage, or current limitations for your project so I don't want to go looking for specific items to recommend just for it to be too big or too power consuming. Good luck!