If National Cartridge Co. is the same as National Arms Co. it was a company in Brooklyn from 1865 to 1870 when Colt purchased them in 1870.
Or it could be St. Louis based National Arms Co that was popular for their .22s and was around from 1908 until being sold to Western in 1909,
Or it could be the Toronto based National Arms Co. that maybe also had a US office… Ask Guy Hildebrand about this company! =)
But you mentioned Air rifles, so it is most likely the company who made this:
You can see National Cartridge Company / Brooklyn on bottom side of tin.
Person who owned these said they were marketed in the late 1950s