In order to deal with missing data, I would like to recommend you : https://towardsdatascience.com/how-to-handle-missing-data-8646b18db0d4

You might be able to estimate if there were sufficient records of streamflow, well levels, and possibly reservoir levels from which to estimate storage changes. These records directly respond to rainfall. It probably would be up to you and others to develop regressions or models to make estimates. If you have heard of flow estimates using the unit hydrograph approach, one might reverse this process to use the gauging station stream hydrographs to estimate rainfall. I have not attempted this, and there can be substantial geospatial variance with individual storms, but perhaps the ability of watersheds to yield a landscape average, maybe this would be good enough to estimate monthly values.