Ultrasonic generator - a couple of questions
Yes, this is the simplest way to do it.

C1 and C2 are needed for supplying current to the transformer as it switches, as its supply is decoupled from the main +5V by a 100Ω resistor. This is done to avoid ripple coming from the transducer circuit leak into the main +5V.

The center-tapped transformer is needed to keep it simple, as you cannot deliver a DC component to the transformer core. Which you would if you used only one transistor. If you don't have a center-tapped transformer, you need a H-Bridge of four transistors, and correct triggering.

About the added questions:

C1 ist needed because C2 is a 220µF electrolytic cap and thus, its ability to deliver sharp spikes of inrush current into the transformer is fairly limited. The internal inductance of such caps is rather high. C1 is a low-inductance type cap which does the job until C2 can deliver.

If you use an external power supply for the transducer circuit, you have to leave out R1 (only connect GND), and you may leave out C2, but not C1. That is because the cable to your power supply has an inductance, too.