The Australian-English phrase the first (or the next, the last, etc.) cab off the rank means:
– the first (or the next, the last, etc.) in line;
– the first (or the next, the last, etc.) in a series of people or things to arrive or appear;
– the first (or the next, the last, etc.) to take advantage of an opportunity.

This phrase refers to cab ranks (i.e., designated areas where taxicabs line up to wait for business), which operate on a first come, first served system in which the cab at the front of the line serves the first customer, the remaining cabs moving ahead as each cab departs.

These are, in chronological order, the earliest occurrences of the phrase the first (or the next, the last, etc.) cab off the rank that I have found:

1-: From the column Secret Service, published in the Sydney Sportsman (Sydney, New South Wales) of Thursday 7^th February 1952:

Trained at Randwick by Harry Darwon, HAYDOCK went around in last Saturday’s Two-Year-Old Maiden for colts at Canterbury with the stable not over confident of success.
And that’s just how it looked soon after the field had settled down. One of the tailenders until they were well on their way to the judge, HAYDOCK was repeatedly cramped for room down the running, yet when Royal Signal, Argentina and Taranta Ra went over the line, he was next cab off the rank.
While we’re not suggesting HAYDOCK will ever turn out the best oat burner in Darwon’s menage, it’s our opinion he’ll develop into a decidedly useful horse.

2-: From The Sporting Globe (Melbourne, Victoria) of Wednesday 1^st July 1953:

“Stake” for Scobie

Victorian jockey “Scobie” Breasley is not troubled by the meat rationing still in vogue in England. Each Friday the choicest cuts are prepared by a Cronulla butcher and placed on an oversea plane, which leaves Mascot the same night.
At 6 a.m. on Monday Breasley is first cab off the rank at Croydon aerodrome in England to meet the plane, and is untroubled for the remainder of the week.

3-: From “Bet London to a brick on”—Ken Howard’s Sydney review ¹, published in The Sporting Globe (Melbourne, Victoria) of Wednesday 19^th August 1953:

Best Epsom trial I have ever called

Sky Hawk’s win in the second Ashbury Graduation Stakes on Saturday was the best Epsom trial I have seen in seventeen years of racing broadcasting. Trainer T. J. Smith was mystified after the race […].
Asked about future plans, Smith said that Sky Hawk would probably continue to start in Graduation Stakes events, but there was a chance that he would run in the Glenlee Handicap over a mile at Warwick Farm next Saturday.
I think that I will be first cab off the rank and take my first Epsom and Metropolitan double, commencing with Sky Hawk of course, before the Warwick Farm meeting. At the moment, Sky Hawk figures on the eighth line in discussions on the big mile, but it won’t be long before he’s elevated sharply.

¹ The Australian horseracing commentator Ken Howard (1913-1976) popularised the phrase London to a brick.

4-: From the column Grandstand View, by Peter Lyons, published in The Canberra Times (Canberra, Australian Capital Territory) of Friday 24^th June 1966:

Balanced contests

According to an American sports magazine, the Americans have virtually solved the problem of unbalanced competition with a system of pooling new players each year.
[…]
In both professional basketball and football officials realised that it was an economic necessity to even the competition because professional sport, more so than amateur competition, relies heavily on attracting large attendances.
The idea of the system is that all new players coming into a competition at the start of a season are pooled by the controlling authorities.
Each club then is allowed to select three players. But the catch is that the first “cab off the rank” is the team which finished at the bottom of the ladder at the end of the previous season and so it works up the ladder with the premiers taking its turn at the end of proceedings.

5-: From Must we butter up the dairy farmers?, published in The Bulletin (Sydney, New South Wales) of Saturday 22^nd July 1967:

Within Australia, private enterprise factories are expanding sales of yoghurt, cream cheese, and other products on the home market. There is a high degree of American investment in private-enterprise dairy processing in Australia and this is now offsetting the “orderly marketing” system which grew out of the depression and which was consolidated in the Chifley era ². Meanwhile the co-operatives are merging themselves into monoliths. The three biggest co-operatives in Victoria are merging this month into Amalgamated Co-Operative Marketers Ltd. and in N.S.W., where the number of co-ops has fallen from 142 to 88 in the past 20 years, there is fear—for good reason—of intrusion into their territory by both private enterprise and giant Victorian co-ops. It is believed that a butterfat quota system, administered by the Dairy Board, would lead to a return to “orderly marketing.”
It might also, of course, deprive the consumer of dairy products he is entitled to have—just as he is entitled to margarine from wholly produced Australian oilseed. But the history of “orderly marketing” in Australia suggests that the consumer is the last cab off the rank.

² This refers to the Labor statesman Joseph Benedict Chifley (1885-1951), Prime Minister of Australia from 1945 to 1949.