Quite simply, you won't.
You have two main problems.
Bullets are just pieces of metal. They might even be highly engineered pieces of metal (or even ceramics) depending on the round. What a weapon shoots is a round: the combination of casing, propellant (gun powder), bullet, and primer (which ignites the powder).
A syringe gun doesn't have those components, because a "dart" is not a bullet. It is not a "dumb" projectile meant to penetrate, or otherwise injure the target. Therefore its a lot more fragile, it can't take the forces which are exerted on the bullet.
2) Aerodynamics and force
When your dart slams into the target it must achieve 3 things:
- Be aerodynamically stable and hit the target
- Deliver its payload into the target's body successfully
- Not hit the target so hard that it penetrates it (like a bullet does), or so weakly that the payload is not injected
But darts, as we've established, are not bullets. They are not nearly as aerodynamic. And they carry no propellant (not that they could survive the pressure generated by igniting a serious amount of gunpowder), therefore they must be fired using compressed gas.
And so, right off the bat, you have 2 major impediments to shooting a target at 400 meters. Shooting a dart full of liquid 400 meters out would require a heck of a lot of pressure and force. So much, in fact, that the dart probably won't survive, and such a gun would require some serious pressure tanks to operate off of.
Second, darts are fired out of smooth-bore guns, they are not made to be shot out of rifled barrels. This all means that they will be way more aerodynamically unstable than a bullet. Hitting something at 400 meters is essentially a pipe dream (pun intended).
Dart guns are meant to deliver a payload at short to medium ranges, and must typically be well aimed, as the dart should not impact the target in the eye, for example. What you're looking to make is some sort of dart sniper rifle, and that won't work.