How to handle a ncdf raster in R

I'm working with a raster layer that looks like this:

class      : RasterLayer 
dimensions : 7040, 9020, 63500800  (nrow, ncol, ncell)
resolution : 1, 1  (x, y)
extent     : 0.5, 9020.5, 0.5, 7040.5  (xmin, xmax, ymin, ymax)
crs        : +proj=longlat +datum=WGS84 +no_defs 
source     : omi_surface_no2_2005.nc 
names      : surface_no2_ppb 
zvar       : surface_no2_ppb 

And I need to re-project it so that I can crop it to this shape file:

Simple feature collection with 1 feature and 0 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: -80.24869 ymin: 43.05051 xmax: -78.43696 ymax: 44.51657
Geodetic CRS:  NAD83
                    geometry
1 MULTIPOLYGON (((-79.38536 4...

But when I try to re-project it, it doesn't work.

This is the code that I used:

omi_reproj <- projectRaster(omi, crs = crs(gtha_shp))

Has the raster layer been assigned the CRS incorrectly? I'm not sure what I'm doing wrong. If you need the data layers, I can provide a link.

Link to raster: link

This file does not follow the CF conventions for storing raster data.

library(terra)
#terra 1.7.11
x <- rast("omi_surface_no2_2005.nc")
#[1] "vobjtovarid4: **** WARNING **** I was asked to get a varid for dimension named dim1 BUT this dimension HAS NO DIMVAR! Code will probably fail at this point"
#Warning messages:
#1: In min(rs) : no non-missing arguments to min; returning Inf
#2: In min(rs) : no non-missing arguments to min; returning Inf
#3: In max(rs) : no non-missing arguments to max; returning -Inf
#4: [rast] cells are not equally spaced; extent is not defined 

You can still read the values with the ncdf4 package and create a SpatRaster "by hand".

Open the ncdf file and inspect its contents

library(ncdf4)
nc <- nc_open("omi_surface_no2_2005.nc")
nc
#    3 variables (excluding dimension variables):
#        double surface_no2_ppb[lon_dim,lat_dim]   (Chunking: [752,587])  
#        double LON_CENTER[dim1,latdim]   (Chunking: [1,9020])  
#        double LAT_CENTER[londim,dim1]   (Chunking: [7040,1])  

Read the relevant data and close the file

lon <- ncdf4::ncvar_get(nc, nc$var[["LON_CENTER"]])
lat <- ncdf4::ncvar_get(nc, nc$var[["LAT_CENTER"]])
d <- ncdf4::ncvar_get(nc, nc$var[["surface_no2_ppb"]])
nc_close(nc)

Create a SpatRaster

library(terra)
x <- rast(t(d[, ncol(d):1]), ext=c(range(lon), range(lat)), crs="+proj=lonlat")

plot(sqrt(x))