As @DavidRicherby said, just "rainfall" would be more natural than "rainfall activity."
Sentence 1 – in
The rainfall has gone down in the past two days.
This means that, at some point in the previous two days, the volume of rainfall has decreased. There is no indication of whether it decreased gradually over the two days or not.
Sentence 2 – over
The rainfall has gone down over the past two days.
This sentence has at least two meanings that I can see:
That the amount of rain that is falling has, over the past two days, decreased; or
That, in comparison to the volume of rain that fell in some other time frame, the volume of rain that fell over the past two days was less.
Here's an example for the second meaning:
On Monday and Tuesday we measured 15mm of rainfall. On Wednesday and Thursday we measured 10mm of rainfall. Today is Friday, so the rainfall has gone down over the past two days.
Sentence 3 – for
The rainfall has gone down for the past two days.
This has a similar meaning, but to me it has different connotations. It means that the volume of rainfall has gradually decreased, but it also implies that multiple measurements were made over the course of the two days and at each measurement the volume had decreased.
Sentence 4 – during
The rainfall has gone down during the past two days.
This sentence doesn't sound natural to me. It would certainly be understood, but it doesn't sound natural. However, this may just be a dialect difference between British and American English. If we use Google Ngram Viewer, we can see that "during the past two" used to be significantly more common in American English than it was in British English:
Using over is the safe option, since in both cases during is declining in usage. However, it does appear that during is correct.