Apologies if you already know some of this but it's easier to start with the beginning. An ultrasonic probe (5-10MHz) is excited with a high voltage pulse which sends out a short pulse of mechanical movement in the compression mode. This hits something and is reflected. The probe receives this reflection after a time and converts it back to a small voltage. This received signal will need to be amplified to reliably convert it into a measurement. Once amplified the magnitude and time of arrival after the pulse was transmitted is captured by digitising the signal with an ADC. There are plenty of lower frequency (40Hz) air coupled distance sensors for the uno available. I would think that the code can be adapted eg https://create.arduino.cc/projecthub/abdularbi17/ultrasonic-sensor-hc-sr04-with-arduino-tutorial-327ff6

The frequency needs to be high (MHz) to give any hope of measuring the thickness unless you do some complex maths and can resolve the thickness by a function of resonance. Wavelength in the material should be function of thickness at around 10 times less. with the average pipe wall being say 5 mm.

I hope this gives you a few things to think about and helps you with the project.

I guess you need to do the sending and receiving in the same position. The compressional soundspeed in metal is high, 3000-6000 m/s, so you need a quick setup, and resolution.

I have a collegue who made his PhD on ultrasonic testing of pipes for corrosion. That was a more complex system with a ring of piezoelectric transducers mounted around the pipe, both seding and listening to the weak reflected pulse. A pipe is a shell structure, so the fundamental pipe wall frequencies are thickness independant. At least true for the ring mode.

From my point of view there is no chance to use Arduino Uno for this purpose. You need short pulse transducer (contact or immersion) with probably at least 5 MHz main frequency and ADC converter with at least 25 MHz. Additionally, you must be able to excite such transducer with some realistic pulse. Probably at least 20 V and 100 ns length. You will also need an amplifier with suitable bandwidth.