Water heater noises when my neighbor's sprinklers are active
Oops tried to edit and deleted. 
To me it sounds like you are hearing the pressure changes when his system kicks on your main pressure may drop, as you said no tank you hear more and it could even be the controls in the tankless that tanked water heaters do not have. You might want to add a regulator or pressure reducing valve on your main incoming line and reduce your pressure 5-10 psi this should be enough to prevent noise caused from main pressure fluctuations, if you think he has tapped your line when his sprinklers start go out and shut off your main to the house at the meter. If his sprinklers keep going it is just the pressure change. I doubt you need an expansion tank with a tankless as all the ones I have installed require a flow to heat, a regulator set at 70psi should eliminate the noise caused by pressure shift since the pressure is 75 when he is running the sprinklers.

Added because comment was two long , have you ever had the water temp in the shower change when someone flushed or turned on the water? A regulator on the supply lines to the shower valve(s) can eliminate this it’s the same principal the regulator set below the minimum pressure eliminates the change. I think a flow in your main supply is causing a pressure fluctuation that you can hear because of the type of water heater you now have has controls that are pressure sensitive flow or static. Eliminate the changing pressure and no more noise.