what is the best way to detect spatial and temporal pattern of precipitation in mountainous region?
This is a non-trivial problem.  Short answer:  I don't think it can be done with monthly satellite data to a useful precision.

Here's where I started:

https://earthobservatory.nasa.gov/Search/index.php?hq=site%3Aearthobservatory.nasa.gov%2FFeatures%2F&amp;q=measuring+rainfall

They now have a tropical rainfall project.  

https://earthobservatory.nasa.gov/Features/TRMM/

It uses space borne precip radar.  In the article above I didn't find specs on the resolution.  Looking elsewhere, it's temporal resolution is 1 month, and horizontal is .25 degree (about 18 miles) If your study zone has latitude under 35 degrees, it may be of use, but overall it seems too coarse.  I would expect a montane environment would require a resolution on the order of a kilometre. 

Even a simple rain gauge is problematic.  Go look at the siting requirements.  I suspect that parking one on a cliff at 6700m is going to give you non-conformant data.

I think synthetic aperture radar is being used to determine snowpack depth. 
In the bad old days they hired people to go measure snow packs.  

Using corner reflectors on the surface where they would be buried in snow may give you better results.  You measure the attenuation from the snow on top of them.  Take substantial time and effort to calibrate. Big win:  You are measuring the mass of ice in the column, not the depth.

Rainfall is much more difficult.  Water doesn't pile up and wait to be measured. 
Or maybe you could stack it:  Consider a corner reflector at the bottom of a cone.  Choose a frequency that is somewhat absorbed by liquid water.  The cone is deliberately angled so that reflections off of it miss the transmitter.  So the corner reflector's signal is attenuated by the pile of water on it.  The cone magnifieds the water depth.

Weather radar can determine precipitation intensity, but using it in the mountains would mean getting returns of the mountain too.  Hard to sort. 

Possibly put the radar on top of a mountain, and give it zero angle of elevation. In essence you would be measuring the amount of water passing through a disk. I don't think these instruments come in crackerjack boxes.

However, you said satellite.

If you have known chokepoints in stream channels, you can monitor them using very high resolution imagery -- in effect you are using a natural weir.  However you need resolution in inches. I'm skeptical about the folks at Langley lending you their satellites.

Might be possible to monitor the rise and fall of lake and ponds.  This requires a precision that I don't think is possible right now.

Might be possible using a pair of corner reflectors.  One is fixed to the ground, the other floats in a plastic open top barrel.  Bounce microwave pulse and measure the difference in time.  This requires accurate timing.  A nanosecond is about 9 inches at speed of light.  You want to reliable measure 1/10 of this.  Centimeter band microwave should be able to do this.  You still have to place the instruments, and I suspect you will need to visit them to clean out dead leaves, bird nests, and replace instruments that have been damaged by moose and woodpeckers.  You will also need to do calibration measures to compare what the instrument in the open measures compared to ones under the tree canopy.  

Your upper levels are well above tree line, which means that as often as not, precipitation travels horizontally.