a) A fresh of breath air
Lenehan has to sing for his supper, and he does so with a stock of mostly music-hall, pantomime witticisms and popular lore. Here is one ('a fresh of breath air' rather than 'a breath of fresh air') which he has picked up from sources such as the one cited below, and to which he gives another airing in Ulysses:
One of his [Leslie’s] remembered witticisms in the pantomime was the exclamation, 'Let us go and have a fresh of breath air,' upon which sentence he rang a variety of changes.1
William Thomas Vincent, Recollections of Fred Leslie (1894) vol. 1, p. 31
b) Catching a cold through an open gate
The following quip about the cold picked up in the park because the gate was open had some currency in a slightly different wording for almost a century:
"What's the matter with the young gentleman?" growled Mr. Gravel,
with his eyes wide open. "He has caught a cold," said Miss Daisey, with a deep sigh. "Ah!" growled Gravel, "slept in a field last night, and left the gate open, I suppose, eh?"
Fraser's Magazine (1838), November, p. 606/2
Jack: You've got a bad cold, Pete.
Jack' How'd you get it?
Pete: I slept in a field last night and someone left the gate open!
Boys' Life (1920), July, p. 46/3
It seems possible that Lenehan’s insinuating remark about Molly Bloom was picked up as a unit that comprises fresh air and the open gate joke. In 1908 Dr Woods Hutchinson A.M., M.D. published a report promoting the benefits of fresh air. The Medical World approved, advising:
Therefore get the fresh air habit. It may now be asserted that sleeping out- of-doors or in the open air is feasible and beneficial in all climates and in all seasons. (Volume 26, p. 444)
The newspapers eagerly picked up the slogan “Get the fresh-air habit”. The article by Dr Hutchinson (who became well known for his views on the ventilation of public places), was widely syndicated:
Get the Fresh-Air Habit.
You can't catch cold by sitting in a field exposed to the draft from an open gate. Though I understand that casuists of the old school of "the chill-and-damp" theory of colds are still discussing the case of the patient who "caught his death o' cold" by having his gruel served in a damp basin.
Saturday Evening Post (1908), 11 March, p. 11