PUMP SELECTION is your main concern.
if you need "puff's" of a mist, then a constant pressure and a fuel injector might be one way.
it appears that amortization of the fluids is purpose and I do not think a peristaltic pump is useful in any way. the pump is not continuous, it pulses, and the pressures created would not be high enough to create amortization.
or... a calibrated chamber volume in the pump might deliver the exact volume required....
this is all unknown, no data from which to draw information.
I would think a piston might.
Since water is a non-compressible fluid...you would need to compress it in a chamber.
a sealed chamber, with a piston exerting X force,... (Charles Law)
a valve opened, would alter the fixed container, and the pressure would then reach equilibrium with the outside pressure by carrying the fluids. A fixed pressure would deliver a fixed volume.....
I would think a high pressure vessel with a solenoid might allow the high pressure to be exposed to the orifice for definite periods of time like in a fuel injector.
a low pressure, micro-atmospheres, would carry vapor so that there would be no impact due to the speed of the delivery based on changing pressures.....
But pump selection is more mechanical selection than Arduino related. And the amount of information given is not making it easy to help in that area.
So, in answer to your question : YES the Arduino can control the to the timing you listed.
You can set that time, even have a menu or knob to alter times, duration, etc.
with a display, etc.
what you are controlling is uncertain, and if it can respond to the inputs is unknown.
kinda like ... just because the land speed record of of 763 MPH was set by a car... and I have a car....