no
The lit sign in the studio usually says "on air" but that's an abbreviation. Anyway, Google finds 801 instances of "put him on the air" around the net, but only 63 of "put him on air". Some of the latter may refer to literal air (as in, "put him on oxygen") and taking care of someone unable to breath. But some others do seem to mean the same as "on the air", so I guess you could use either phrasing.
