There are many solutions for such application, each solution will depend on the requirements of your system:
► You can use DC-DC Converter Charge Pump. Charge pump step up voltage is able to take (in) lower voltage to supply (out)higher voltage. As an example this charge pump (EXR Mfr Part No: SP6661EU-L) is able to double the supply voltage, if you apply on it 5(V), it will be able to generate 10(V).
► Buy a cheap PC Power Supply to power the fan. Power supplies convert AC input to lower DC voltages. The typical voltages supplied are: 3.3(V), 5(V), and 12(V). The 3.3- and 5-volts are typically used by digital circuits, while the 12-volt is used to run motors in disk drives and fans. There are many wires inside the PC power supply case, the yellow wire is 12(V), Red 5(V), Orange 3.3(V), Black Ground, White -5(V). For your application you can power the DC fan by the yellow and the black wires from the PC power supply.
►You can use a Solar Panel (photovoltaic). The solar panel has two terminals - positive and ground -
You can use batteries, you can charge capacitors then use them to power your circuit. There are many solutions each one of them is depending on your intended use.
No, you cannot use that supply to run that fan @ 12 volts…don’t forget watts law.
The Fan = 12V X .24A = 2.88 watts
Supply = 4.74V X .2A = .95 watts
Forget about trying to raise the voltage on that supply, it probably won’t supply the power needed.
Forget about buying a PC power supply to drive one little fan. LOL
Forget about the solar cells and the battery…
And the caps…lol…A bazillion UF cap would run that fan for about the time you can sneeze.
Buy yourself a small regulated 12 Volt DC wall wart, and save yourself a great deal of time and trouble.