Your data is completely non-standard and not self-descriptive. In other words, no library will give you a one-stop solution.

With package ncdfCF I see the following file structure:

> library(ncdfCF)

> irr <- open_ncdf("~/withd_irr_h08.nc")
> irr
<Dataset> withd_irr_h08 
Resource   : withd_irr_h08.nc 
Format     : classic 
Conventions: (not indicated) 
Keep open  : FALSE 

Variables:
 name      units    data_type axes           
 withd_irr mm/month NC_DOUBLE grid_num, month
 lon       degree   NC_DOUBLE grid_num       
 lat       degree   NC_DOUBLE grid_num       

Axes:
 id axis name     length values                      unit               
 1  T    month      480  [1971-02-01 ... 2011-01-01] months since 1971-1
 0       grid_num 64028  [1 ... 64028]

The data variable withd_irr does not use a longitude-latitude grid, instead it has an axis grid_num which holds 64,028 locations, same as the lon and lat variables (which you want to be axes if terra should recognise them, but they are not).

The solution to your problem is to extract the data for the 3 variables, put them in a data.frame, then convert that to a terra::SpatRaster. You will have to do the conversion for each month individually because the various steps are available for matrices but not for arrays:

# Read entire data arrays
> lon <- irr[["lon"]][]
> lat <- irr[["lat"]][]
> withd <- irr[["withd_irr"]][]

# Make a data.frame with lon, lat and all 480 months of withd_irr
> df <- data.frame(lon, lat, withd, check.names = FALSE)

# Convert a month of data to a SpatRaster
> r <- rast(df[c("lon", "lat", "1971-02-01")])
> r
class       : SpatRaster 
dimensions  : 279, 720, 1  (nrow, ncol, nlyr)
resolution  : 0.5, 0.5  (x, y)
extent      : -180, 180, -56, 83.5  (xmin, xmax, ymin, ymax)
coord. ref. :  
source(s)   : memory
name        : 1971-02-01 
min value   :    0.00000 
max value   :   96.28087

All months in one go

Putting it all together and making a single SpatRaster:

> library(ncdfCF)

> irr <- open_ncdf("~/withd_irr_h08.nc")
> lon <- irr[["lon"]][]
> lat <- irr[["lat"]][]
> withd <- irr[["withd_irr"]][]
> df <- data.frame(lon, lat, withd, check.names = FALSE)
> mon_str <- names(df)[-(1:2)] # drop lon, lat

# Build the SpatRaster structure from the first month
> r <- rast(df[c("lon", "lat", mon_str[1])])

# Now add the remaining months
> for (i in 2:length(mon_str)) add(r) <- rast(df[c("lon", "lat", mon_str[i])])
> r
class       : SpatRaster 
dimensions  : 279, 720, 480  (nrow, ncol, nlyr)
resolution  : 0.5, 0.5  (x, y)
extent      : -180, 180, -56, 83.5  (xmin, xmax, ymin, ymax)
coord. ref. :  
source(s)   : memory
names       : 1971-02-01, 1971-03-01, 1971-04-01, 1971-05-01, 1971-06-01, 1971-07-01, ... 
min values  :    0.00000,     0.0000,     0.0000,     0.0000,     0.0000,      0.000, ... 
max values  :   96.28087,   150.5525,   491.3734,   448.1106,   269.3665,    331.089, ...