Do DSP principles apply in the same way to both Audio and EM spectrums?
Many of the ideas will be similar, but some assumptions may no longer apply. For instance many signals in EM can be treated as Narrowband (narrow bandwidth with respect to the centre frequency), in Ultrasound you may run into Broadband signals more easily. Another problem you may run into is the speed of propagation varies as a function of frequency.

Exactly what difference you encounter will depend on your experiences with EM and what exactly your are doing in ultrasound.