My bachelor work was an ultrasonic anemometer.
I used an ultrasonic transducer (400ST160), 4 receivers (400SR160) and a microprocessor (STM32F100RBT6B). The aim was to calculate the time of flight (TOF) of ultrasonic signals transmitted from the transducer to the receivers.
The STM32 timers make things easy with the complementary output and dead time feature. I generated a chirp signal (i.e. a signal with increasing frequency) to cover the transducer's and receivers' frequency band. The complementary outputs go to MOSFETs (IRF7103PBF) which drive a transformer (ER11-3E5-S) with a central tap to generate a dual polarity signal.
The transformer generated a high voltage (+/- 90V), because ultrasonic signals decay like R^2 in the air. The RX circuit is also simple. The received signal goes through protective diodes and filters to the amplifier (AD8544ARUZ 4 in one) to amplify small signals which go to ADC channels of the STM32. The STM32 has a 12-bit 1MSPS ADC. I calculated the correlation between the RX and TX signals to estimate the TOF, but use whatever algorithm suits you best.
I recommend using dual polarity signals. If it is too hard to generate chirp signals, simply generate burst signals, and search for peeks in the received signal. Those mentioned methods have benefits only if you need accuracy. The main problem with these transducers that they are narrowband like yours.