Have you tried the http://www.worldclim.org ? The 30'' datasets corespnd to 1km on Equator.

There are some precipitation variables there (Annual Precipitation, Precipitation of Wettest Month, Precipitation of Wettest Quarter, and so on) but if you look on the databases, you could find the average monthly precipitation as well 

These are not on satellite-based measurements but based on (a few) point measurement on the ground interpolated by lat, long and elevation and therefore of unknown accuracy.

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Hello Onur,

the site to visit is that of the International Precipitation Working Group (IPWG):

http://www.isac.cnr.it/~ipwg/

In particular, you want to go to the page of the datasets:

http://www.isac.cnr.it/~ipwg/data/datasets.html

Cheers

check TRMM & GPM Precipitation Data here:

http://pmm.nasa.gov/data-access

https://knoema.com/icrqbyg/monthly-rainfall-data-district-wise-for-2004-2013?action=download

 http://coastwatch.pfeg.noaa.gov/erddap/info/index.html?page=1&itemsPerPage=1000

http://earthexplorer.usgs.gov/

The Nasa World Observations 

http://neo.sci.gsfc.nasa.gov/view.php?datasetId=TRMM_3B43M&year=2015

Follow as said by Kleomenis Kalogeropoulos for TRMM and GPM data

Here is a daily 1 km rainfall data for Africa.

http://www.cpc.ncep.noaa.gov/products/fews/AFR_CLIM/arc2_201303_final.pdf