OK, Xander, caipirinhas it is.
Yes you are right, the municipalities are pretty small, although some are huge. I'm annexing the layer file with the municipalities.
So far I figured out I can use the Sample—Help | ArcGIS for Desktop tool if with centroid points made of each polygon. The data seems to be fine, and one option would be to get one raster for each statistics (either that or the 12 months and I can do the statistics later), use Sample with points and this set of rasters, and then repeat that for each year. Too bad though the Sample command doesn't let me capture the code of each municipality (variable CD_GEOCMU in the point layer). Alternatively to the centroid point, I can get the coordinates from each city town hall, which are made available by the Brazilian Institute of Geography and Statistics.
But I think ideally, because of the larger municipalities, getting the data from rasters to polygons would be better. To that end I tried the Extract Values To Table—Help | ArcGIS Desktop tool, but the data doesn't seem to fit. One municipality is inside a raster with a ~110 mean precipitation and it comes out with ~142 in the table using Extract Values to Table. I'm checking it with the OID* and FID* identifiers from both tables, though I'm not sure they are exactly the same and I could be seeing another municipality (ideally I would like to CD_GEOCMU be used as the identifier variable...). I'm wondering how these tools weight the data though - if a raster cover 40% of the polygon and another covers 60%, will the mean precipitation be the weighted mean of these two rasters?
So these were the two things I tried so far, Sample worked alright for points, EVTT didn't for polygons. The first-best scenario would be getting it from polygons due to big municipalities.
Also, for the data, I'm not sure it is available on a yearly basis - I'm using this dataset because I saw it being used in an economics paper concerning rainfall and child birth in the Brazilian semi-arid. Also, the precipitation variables are to be used as controls in a regression, so they don't need to be exactly perfect, just need to show the heterogeneity in rainfall between municipalities. As I use panel data, that's why I need data on the 19 years.
I'm sending a .lyr file since the .shp was too big to annex.