The phrase like a rat up a drainpipe (originally, in Australian English, like a rat up a pump) means: very quickly; also: very energetically.
For example, Dominic White used this phrase punningly in the column The Market, published in The Daily Telegraph (London, England) of Saturday 2nd April 2005 [No. 46,597, page 29, column 1]—Rentokil Initial is a British pest-control business:
RODENT catching was in fashion in the City yesterday as traders made for Rentokil Initial like rats up a drainpipe. Credit Suisse First Boston told clients to buy the shares ahead of the new chief executive’s arrival on Monday morning, when former Aegis chief Doug Flynn gets his security pass for Rentokil HQ.
The earliest occurrences that I have found of like a rat up a pump and like a rat up a drainpipe are as follows, in chronological order:
1-: From an account of the sporting events organised by the Mount Gambier Football Athletic Sports on Wednesday 5th October 1881, published in The Narracoorte Herald (Naracoorte, South Australia, Australia) of Friday 7th October 1881 [Vol. 6, No. 396, page 2, column 5]:
The “Obstacle Race,” created much merriment, and was well carried out. […] On a start being made fourteen competitors stormed the six feet fences of the show yards, and it was highly amusing to see the runner [sic] climbing the fences. Some of them went over them “Like a rat up a pump.”
2-: From the Queensland Figaro (Brisbane, Queensland, Australia) of Saturday 3rd February 1883 [Vol. ?, No. ?, page 73, column 3]:
PUNISHMENT follows as closely on crime as the shadow does the substance. Moral. Never “shanghai” your stable helps into the position of Rue de St. Honoré waiters. Not long ago an Ipswich hostess “gif a barty,” and there was no less than one real live Minister at it. All went as slick as a fat boy falling down a cellar, or a rat climbing up a pump until Ostler ’Arry waltzed along with a plate of rissoles. “Deah me!” exclaimed the lady (and the membah sat near her right elbow), “whatever are these, Enrico?” “Well, mum,” replied the unsophisticated straw-chewer, “Mister Johnsing sed as how thrippence was the usual thing, but you was such a dam bad mark he’d hev t’ mek ’em fourpence.” Flashes of silence enlivened the festivities after that lot for half-an-hour, and then the social atmosphere grew as still as a pot of striped paint.
3-: From Sydney Punch (Sydney, New South Wales, Australia) of Saturday 20th October 1883 [Vol. ?, No. ?, page 428, column 2]:
The Ruling Passion.
SALVATION Harmy Scene, fashionable part of Collingwood. Burglar Bill (recently converted, and now first trombone in the band) to Sammy the Sneak, another Capting: “Sammy, them’s a lot o’ tidy cribs up there—must hold lots o’ plunder.” “Looks like it Willum, it does,” said Sam. Bill: “Yes, an’ nary dawg t’ be seen, an’ then, Sammy, twig them there little bal-co-nies—why a feller cud slip up one o’ thim like a rat up a pump; then he teks a pane out o’ the glass door nice an’ soft, an’ then”—here William suddenly remembers his conversion, blows a fierce Ta-ran-ta-ra, and drops the subject.
4-: From a correspondence from Penola, in South Australia, published in The Border Watch (Mount Gambier, South Australia, Australia) of Wednesday 23rd April 1884 [Vol. 22, No. 2,135, page 2, column 7]:
Some twenty, more or less, wended their way to MacDonnell Bay to have a header in the C. Man proposes frequently, however, only to be disappointed, for on arriving at the bathing-house they found the door was locked! […] Suddenly a small voice arose from the midst of the crowd, and plaintively warbled, “If you can’t get in at the golden gate get over the garden wall.” The effect of the lad’s suggestion was electrical! In a twinkling he was hoisted on to the shoulders of stalwart “Harry,” and that small boy felt as “’appy as if he was in his huncle’s harms” while he went up the wall like “a rat up a pump.”
5-: From a special dispatch from Columbus, Ohio, dated Saturday 30th January 1909, published in The Cincinnati Enquirer (Cincinnati, Ohio, USA) of Sunday 31st January 1909 [Vol. 46, No. 31, page 6, column 2]:
When the senatorial fight opened there appeared a cleavage in this band of brothers. Some of the old guard, Sherman Eagle, John R. Molloy, Warden Orrin B. Gould, A. R. Critchfield, Newton Miller and others too numerous to mention in this limited space went through the crack like a scared rat up a drain pipe.
6-: From Chapter XIV of In Fear of a Throne, by R. Andom and R. Hodder, published in The New Zealand Herald (Auckland, Auckland, New Zealand) of Wednesday 1st September 1909 [Vol. 46, No. 14,154, page 10, column 1]:
“If there’s anything clear to my mind,” said the Yankee, “it’s this: we needn’t expect to hear from Wolfram while he’s holding Trod. He thinks he’s got the key to the sit., and you bet he’s turned it in the door, and is waiting on the time when old King Cole pegs out. Wolf. knows that the Prime can’t go on with the game without Trod.; so he lies low, like Br’er Rabbit, and goes on saying nuffin’.”
[…]
“Supposing old King Cole does peg out?” said I, in answer to his remark.
“That would make a swiping difference,” he returned. Then, plastering his right forefinger on to his left thumb, he continued: “In the first place, Wolf would be after Karl like a rat up a pump. Beer-pots would go to glory. Flies would remain on the wall.”
7-: From The Jinx, in The Jinx: Stories of the Diamond (New York: G. W. Dillingham, 1911), by the U.S. sports writer Allen Sangree (c. 1878-1924) [page 46]:
Manager McNabb was last to bed. “It’s a road team, this is,” the men heard him say; “I want seventeen games—we’ll go through ’em like a rat up a pump.”
8-: From a letter that one J. Ludgate wrote from German East Africa, published in The Wiltshire Times and Trowbridge Advertiser (Trowbridge, Wiltshire, England) of Saturday 22nd July 1916 [Vol. 63, No. 3,352, page 3, column 1]:
You would laugh to see them slaughtering bullocks here. The butchers are natives, and they get the herd in a bunch and slip a rope round the hind leg of one of them. They then hang in, keeping the one hind leg off the ground and push it over on its side and when down they rope him up and the butcher cuts its head right off. But sometimes the slip comes off the leg and then you see a sight. The bullock goes after the fellows and they slip up trees like a rat up a pump.
9-: From an account of how Mr. J. W. Ramm, head cutter in Grace Bros.’ tailoring department, recaptured one of the Germans who had escaped from the Victoria Barracks—account published in The Sun (Sydney, New South Wales, Australia) of Friday 12th April 1918 [No. 2,428, page 7, column 2]:
Mr. Ramm was excited by the recital of his story. “That German,” said he, “pulled up as if he had been shot. ‘Mister,’ he said, ‘I will go quietly.’ ‘If you don’t,’ I said, ‘you’ll get it quick and lively.’ ‘I surrender,’ he said, ‘don’t shoot!’
“Well,” said Mr. Ramm, “I couldn’t very well have shot him with a pair of tailor’s shears even if I had wanted to. I handed him over, and I’m after the rest like a rat up a pump, but they had disappeared into lanes and houses near-by.”
10-: From Jack Ozar, Champion Wrestler, Joins the State Police Force, published in the Harrisburg Telegraph (Harrisburg, Pennsylvania, USA) of Saturday 19th October 1918 [Vol. 37, No. 233, page 9, column 4]:
Jack went through the physical examinations like a rat up a pump; nearly broke all the mechanical devices for registering. He also passed a blue-ribbon mental examination.
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