I think this should work...
The Month field should actually follow the post fix of the name to indicate which month it referres to, but I suppose the field would not even be necessary (at least not for the extraction of the raster values to the points).
You could load the mosaic 12 times and specify a different definition query for each mosaic in the TOC, like:
Name = '%ppt1'
This would filter all the rasters to show only those that refer to January (if naming is consistent).
If you would run the Extract Multi Values to Points tool with these 12 mosaics, you would be able to get the 12 dates for all the points. Be sure to change the names of the layers to reflect the month of the definition query, since the tool will show these names and it will allow you to easy define the output field for each raster (month).