I agree with Dana, a micro would be far better suited to this task. It doesn't have to be Arduino, almost any MCU can do it.
To do it purely in hardware you need an oscillator to produce the cycles within the burst, a timer (or monostable) to decide the length of the burst and an oscillator rapidly clocking a counter to find the time before the echo returned. There would also be some logic to coordinate these processes and read the the counter back so you can use it's value.
Yes, the processor-less design involves basically a master oscillator, periodic trigger generation and time intervall measurement between trigger and echo pulse output. Just a handfull of CMOS logic devices.
Another block language you can use, mBlock 3.0. Note once you are
finished with blocks the Arduino code is generated for you as shown
in right hand window. Eg. mBlock generates the code from your block
layup.