Looking at the netcdf file with ncdump -h cons_irr_h08.nc indicates the dimensions of the file are 480 months by 64028 grid_num, with a level of abstraction from grid_num into lat, lon. The data is structured as a set of 64028 individual time series. I don't think this will easily parse into the raster package.
$ ncdump -h cons_irr_h08.nc
netcdf cons_irr_h08 {
dimensions:
grid_num = 64028 ;
month = 480 ;
variables:
double cons_irr(month, grid_num) ;
cons_irr:units = "mm/month" ;
cons_irr:description = "1. gridded irrigation consumption results: 64028 rows, 480 Months;2.the results is based on the outputs of H08 forcing by WFDEI" ;
double lon(grid_num) ;
lon:units = "degree" ;
double lat(grid_num) ;
lat:units = "degree" ;
double month(month) ;
month:units = "months since 1971-1" ;
}
Maybe use library(ncdf4) or library(RNetCDF) to deal with file more directly and consider it as spatial point data.
ETA:
Looking at the lat and lon variables with library(ncdf4) shows it is pretty sparse and you need some sort of interpolation tool to re-grid the data to a raster.
library(ncdf4)
ncin <- nc_open("cons_irr_h08.nc")
lon= ncvar_get(ncin,'lon')
lat= ncvar_get(ncin,'lat')
head(cbind(lat,lon))
lat lon
[1,] 32.75 60.75
[2,] 33.25 60.75
[3,] 33.75 60.75
[4,] 34.25 60.75
[5,] 29.75 61.25
[6,] 30.25 61.25
plot(lon,lat,type='p',pch='.')

The "remap the entire field" section from https://rpubs.com/markpayne/132500 or https://gis.stackexchange.com/a/35322/10229 or How to make RASTER from irregular point data without interpolation might be of some help with this data.
Back to the time variable question, you can extract it directly with the ncdf4 package:
library(ncdf4)
ncin <- nc_open("cons_irr_h08.nc")
tvar= ncvar_get(ncin,'month')
... and then you could turn the numerics into dates with this trick adapted from https://gis.stackexchange.com/a/35322/10229 :
add.months= function(date,n) seq(date, by = paste (n, "months"), length = 2)[2]
times = as.Date(sapply(tvar-1,add.months,date=as.Date('1971-01-15')),origin='1970-01-01')
head(times)
[1] "1971-01-15" "1971-02-15" "1971-03-15" "1971-04-15" "1971-05-15" "1971-06-15"