How can I improve this circuit design?
The large cap (C1) would not normally be needed if you are just driving a simple DC motor. Similarly the flyback diode may not be needed if the motor is the only part being run in the circuit.  

What is the idea to have the current flow fast? In your diagram the 220uF capacitor discharging into a simple DC motor would only power it for a very short time, likely not enough time to do anything useful. To get at least some useful motor run time you might need what is called a super cap. These can have several Farads of capacity. 

The only m-motor that saw related to lego kits are those used for example to slowly spin something like a ferris wheel made of legos. If you are thinking about a catapult that can be slowly pulled back then perhaps your ideal would work for such a system.  However if you want to launch a football at the instant you switch the system on you are going to need a much stronger motor.

Also keep in mind that when using a voltage step up module the input current needs to be higher then the expected output current.  You get the voltage increase but at the expense of a lower output current. For example if you want a 6V output near 500ma you would need to have at least a 1000ma input at 3V). That much current might require a moderate size solar cell array, (you would need to check the supplier's specifications), it may need to be several square inches or more, (plus a very bright light over it).

Your science project likely has safety rules. Might I suggest that you think more in the line of a "Nerf" football throwing system? That would be fairly safe and require much less power. Even with that there could still be some safety concerns related to a catapult's swinging mechanism. 

For the connection to the catapult I would suggest something like a string being slowly wound on the motor shaft or possible a rubber band that is slowly wound tight. The catapult is then operated by cutting the string or by opening a mechanical latch holding back the catapult's lever.

For a final idea: There are descriptions/plans on the internet for building repeating catapults (or roman ballista), that are driven by continued cranking. A DC motor might work well on something like this. Of course only use safe projectiles like foam footballs, (or other "Nerf toys").

Good luck with you science project.