With reference to pup in the sense of a young dog, the colloquial Australian-English phrase the night’s (or the day’s) (only) a pup, and variants, mean the night (or the day) is young, the night (or the day) has scarcely begun.

These are the earliest occurrences of the phrase that I have found, in chronological order:

1-: From Goldfields Gossip, by ‘Lydia’, published in The Mirror (Perth, Western Australia) of Friday 2^nd December 1910:

Quite a regrettable feature of the town lately is the number of abominable surprise parties which have been organised. […] When a number of persons, acting under the disguise of organising a social evening, go to a house, where it is seldom all the party is known, and without any warning proceed to amuse themselves, and annoy the occupants, who through ordinary courtesy have to hide their annoyance, is, to quote the remarks of a slangy male friend, “a little bit over the fence.” […]
What fine times men have. Give the average man a glass, some liquid, and a smoke, and, to quote my friend of above, “The night is only a pup at one g.m. ¹”

¹ Perhaps a shortening of GMT, g.m. is used as a humorous substitute for a.m. or p.m..

2-: From The Daily News (Perth, Western Australia) of Friday 24^th May 1912:

ln Sydney this year 21 schools are playing the Australian game ², as against eleven last year. Certainly it should commend itself to parents as against Rugby, in which game three players have been killed this season—and the day is yet a pup, as one League official frequently observes.

² The Australian game: Australian Rules (football)—cf. aerial ping-pong.

3-: From A Foggy Night in Antwerp, by Henry Lawson, published in The Xmas Bulletin (Sydney, New South Wales) of Saturday 11^th December 1915—John Lawrence has just “got on Belgian soil”:

The night was not even a pup yet; it was broad daylight, being Northern summer. So, to help pass away the time till dark, Jack hired a hurdy-gurdy from a tubby little Belgian woman and played some tunes.

4-: From The Black Opal (London: William Heinemann, 1921), by the Australian author Katharine Susannah Prichard (1884-1969):

Paul sometimes moaned about the chances she was missing, and that she could be singing in theatres to great audiences. Sophie herself laughed at him. She was quite content with the Ridge, it seemed, and to sing to people on their verandas in the summer evenings or round the fires in the winter. She might have had greater and finer audiences, the Ridge folk said, but she could not have had more appreciative ones.
If she was singing in the town, Michael always went to bring her home, and he was as pleased as Sophie to hear people say:
“You’re not taking her away yet, Michael? The night’s a pup!” or, “Another . . . just one more song, Sophie!”

5-: From Truth (Brisbane, Queensland) of Sunday 24^th December 1922:

In the Tailor’s Shop.
Cutter Le Gros Lambasted
By a Lady who said He Wrecked Her Life.

The hands of the clock pointed to 20 minutes past the hour of 8. The day was but a pup, and Rowell’s high-class tailoring establishment was a scene of bustle and animation.

6-: From a correspondence from Melbourne, Victoria, published in The News (Adelaide, South Australia) of Wednesday 23^rd July 1924:

Coburg, one of our industrial suburbs, is puffed up with pride. The city has acquired a new town hall. It puts some of the others in the shade. Quite a Continental air about the approach, set out with well-kept lawns and sweeping paths. There was a civic ball there the other night. The official lancers ran into double sets, the town clerks from all the other suburbs being roped in for the exhibition. By the time this capering was over the clock registered 11, and those who had got past the stage of believing that the night was still a pup, made a fractic [sic] rush for the last tram. When will mayors and town clerks get a little wisdom?

7-: From The Northern Champion (Taree, New South Wales) of Saturday 19^th September 1925:

At the Bush Landing.

(By “Johnnie Walker,” in the Australian Forestry Journal):—
For anyone who wishes to study the great timber industry at close range, it is both interesting and instructive to spend a week or two at the “bush landing.” Apart from the insight into the life and habits of the timber worker species gained by a sojourn in the vicinity of the bush-boarding house, one’s knowledge of human nature generally is also considerably augmented. […]
[…]
After breakfast the harnessing of teams, fed while the day was yet a pup, and the great whims with wheels almost as big as the “grande rue” in Paris, swing ponderously out from the landings, the drivers whispering gently to their teams the while.

8-: From Our London Letter, by ‘Wendy’, published in The News (Hobart, Tasmania) of Thursday 3^rd December 1925:

Eleven thirty, and the footlights are behind us. Spanish shawls and gay brocade make a bright patch on the curb as we wait for our taxis or cars.
But it is far too early for bed yet awhile. “The night is but a pup,” we say, so off we go to dance “after the show” at cabaret or night club till two o’clock makes us feel our cosy beds are not to be despised, and taxis once more glide swiftly along the trafficless streets.

9-: From The South-Western News (Busselton, Western Australia) of Friday 24^th September 1926:

THE GAS WORKS
DISCUSS DOGS.

It was only 10 p.m. and far too early to close a meeting which had already been noted for its long-windedness, mainly through the lack of something interesting to discuss. Councillors were content to stay till 1 a.m. if necessary. Cr. Fork-in-hand had drawn attention to the flatness of the meeting and thought it was time to go home.
Cr. Cross-bones: The night is only a pup.
Cr. Streake was suddenly stirred to life.—“Pup? Pup? Ah yes! That reminds me,” said our worthy draper. “I meant to say something tonight about dogs. I have suffered great mental anguish trying to devise a means of saving my drapery from these inconsiderate animals. All nuisances have to be registered—even Ford cars.”
Chorus:—“What would Mr. Shaw say?”
Cr. Streake resuming after this uncalled for interruption, and warming to his subject, raised the council to its feet by saying that Busselton had gone to the dogs.
Cr. Cross-bones: “You mean the dogs have come to Busselton.”
Cr. Peake (amid confusion):—“This is not a time for joking. How many dogs are registered?”
The May-I, Mr. Web-foot restored order by announcing that owing to the heavy run on sausages lately, only 32 dogs were contributing to the council revenue.
Cr. Soda Water: “They are a danger to health.”
Cr. Cross-bones: “The sausages or the dogs.
Cr. Streake (tearfully): “Listen to me, gentlemen.—I—”
Cr. Soda Water: “You must not be frivolous. Can’t you see that our friend Streake is overcome? Can’t you hear his sobs.
Cr. Cross-bones: “That is Great Scott snoring at the Commercial.”
Cr. Fork-in hand: “Will you stop your nonsense.”
Cr. Streake: “West Busselton is to blame. Their dogs are the worst.”
The May-I, Mr. Web-foot, thought the difficulty could be overcome by removing all the telegraph poles and verandah posts to West Busselton.
Cr. Streake: “Oh! You hard hearts. You cruel men of Busselton. I have tried to tell you—.”
Cr. Fork-in-hand (to Cr. Streake tenderly): “Wilst thou have a drink, a drink of water, dearie?”
Cr. Streake: “Y-y-yes p-please.”
Here Mr. Web-foot gave a display of generosity never before surpassed in the history of the “Gas Works.”
“Go, get him a mouthful of lemonade,” he said.
With the May-I’s cupboard open, the meeting became a very “spirited” one.