How to measure ultrasonic sensor without pulsein method?
Rather than give you the straight answer in code form, consider this approach. If you learn how the unit works, you will be better prepared to answer questions presented to you.

The background of an ultrasonic distance measuring device is that it sends out a signal, in the form of sound you cannot hear. The device then listens for any echoes (bounce-back) of that pulse within a certain time period. This time period represents the distance range that the device can register. If the electronics are of poor quality, the minimum distance will be larger than if the device was of higher quality, as the amount of time for the pulse to bounce from a close-in object is going to be very short. The electronics have to be very fast to read such a short period of time.

There are libraries for ultrasonic devices such as the one you note, but that's the "easy way" to use it. If you perform appropriate searches, you will find there are references how to use direct timing for this device, which will satisfy the requirement you have been given.

I could perform the search and provide you a link, but that's half the work. Consider to use terms including your device identifier along with code and timing and you may find useful links. It is also very possible that you will find someone's pre-written code using the timing method.