Sound waves are longitudinal waves of air pressures. What you feel is just the bulk of molecules hitting your hand at different frequencies and positions. The closest feeling I can imagine would be to immerse your hand in an ultrasonic bath. I've done this multiple times as an undergrad, it's nothing special. It feels like your hand is buzzing.

Not quite the same as an ultrasonic bath. This device creates standing waves, so the antinodes (parts of the wave where the amplitude is 0) will not move. I'm pretty sure the waves created by an ultrasonic cleaner are moving. If the wavelength is large enough, you should be able to feel where there is vibration and where there is none.

The locations where the amplitude is minimal (0) are called nodes. The antinodes are where the displacement is maximum. Ultrasonic baths scan through a frequency range so that the standing waves produced by the drivers move around, hopefully covering the whole bath in one scan.

So yeah, the feeling would be similar, but not exactly the same, as you're also in a different medium.