Though it was described as a stone bridge, it was actually a pier, extending out into the shallow waters of Kowloon Bay. It was first built in the 1870s by the Chinese government, enabling easier access to Kowloon City by boat.
The pier changed shape several times, as reclamation ate it up from the landward end, and different extensions were added to it at the seaward end.
It disappeared from sight completely in 1942, during reclamation for the expansion of Kai-Tak airport by the Japanese.
The closure of the airport in 1998 meant there it was possible to excavate the area. This was done in 2008, showing that several sections of the piers were still in place.
Here's a timeline of the main events:
1873-75 - The Lung Tsun Stone Bridge is built from granite
The original pier was built of granite. It measured about 210m long and 2.6m wide, and was laid in the direction of N131-degrees. The works were completed in 1875. [1, item 4.4.2]
At this time, this lay outside of British territory, which only extended as far north up Kowloon as the line of today's Boundary Street.
1892 - Wooden extension added
To reach even deeper water, a timber extension was added. It measured 80m long by 4m wide, and was laid in the direction in the direction of N118-degrees. These extension works were funded by Lok Sin Tong, a local charitable organization of Kowloon City Market, established in the 1880s. [1, item 4.4.4]
1898 - The pier lies within the New Territories
After Britain leased the New Territories from China, the pier was inside British territory. However, the 1898 Convention made two exceptions: Kowloon City would remain under Chinese jurisdiction, and Chinese men-of-war could use the pier. Here is that section of the text :
It is at the same time agreed that within the city of Kowloon the Chinese officials now stationed there shall continue to exercise jurisdiction except so far as may be inconsistent with the military requirements for the defence of Hong Kong. Within the remainder of the newly-leased territory Great Britain shall have sole jurisdiction. Chinese officials and people shall be allowed as heretofore to use the road from Kowloon to Hsinan.
It is further agreed that the existing landing-place near Kowloon city shall be reserved for the convenience of Chinese men-of-war, merchant and passenger vessels, which may come and go and lie there at their pleasure ; and for the convenience of movement of the officials and people within the city.
The status of Kowloon City would be an ongoing point of disagreement between Britain and China over the following years, but I haven't seen any complaints about of a lack of Chinese access to the pier during that time.
1899-1900 - New name: Kowloon City Pier. New timber.
The 1899 and 1900 Annual Reports of the Public Works Department (PWD) noted that the timber section of the pier, which they now called "Kowloon City Pier", had decayed and was reconstructed. 
This 1903 map of the area  shows Kowloon City (G) and the pier (C). The letter C is over the granite section of the pier, then the lower part is the timber section.
1908-10 - Concrete replaces timber
It seems the 1899-1900 'reconstruction' of the timber section was actually a more minor set of repairs, because the PWD's report for 1906  says:
The timber portion of the Kowloon City Pier was however in such a decayed condition that, in view of the fact that it has been decided to replace it with a reinforced concrete structure, no repairs were undertaken.
The solution was to replace the timber section with a new section made from concrete. We can follow the progress in the following years' reports from the PWD:
- 1908: 27 reinforced concrete piles were made for the Kowloon City Pier. 
- 1909: The concrete piles were driven into the seabed, and the area around the pier was dredged. 
- 1910: The concrete pier was completed. It measured 149 feet long, with two sets of steps at the outer end. 
The 1910 report also notes that two launches were allowed to start running a ferry service to and from the new pier.
1920s - Northern part of the granite section is buried under reclamation
Starting from 1916, a new sea wall was built across the northern section of Kowloon Bay, and the area behind it was reclaimed for the Kai Tak Bund project . The line of the new sea wall cut across the granite section of the Kowloon City Pier, so the northern part of the pier was buried under the reclamation. I'm not yet sure which year that happened, but it was likely in the early 1920s.
The southern part of the pier's granite section, and the concrete part, remained in use. The following years' PWD annual reports mention repairs to the pier.
1934 - A new causeway provides easier access to the pier
In early 1934, the government issued a tender for "... the dumping of pell-mell rubble to form a causeway to Kowloon City Pier and the surfacing of same with cement concrete ..." . The work was finished in December of that year , and I believe that is the layout we see in the map below.
So I think we're looking at:
- A: The sea wall and reclaimed land
- B: The southern end of the granite section of the Lung Tsun Stone Bride. (The points of the hexagonal pillars can be seen.)
- C: The 1934 causeway
- D: The concrete pier, either from 1910 or 1937 (see below).
The map collection says the map is from 1922 though, which conflicts with it showing the 1934 causeway. More work is required to confirm exactly when the map was drawn and what it shows.
1936-7 - Reconstruction
In 1936 the government issued another tender , this time for "Reconstruction of Kowloon City Ferry Pier". (Note the addition of "Ferry" to the name.) It says the work comprises: "the reconstruction of the pier in reinforced concrete involving the construction of R. F. C. walings, bracings, landings and steps, decking, roofing, booking offices, barriers and other contingent works."
Before this work began, the pier was made up of the three parts shown in the map above: the southern end of the original granite pier, the concrete pier from 1910, and the causeway from 1934. I'm not sure whether the reconstruction applied to all three parts, or just to that concrete pier from 1910.
The newly reconstructed pier opened in March 1937. 
1942-5 - Lost from sight
During the Japanese occupation of Hong Kong, they reclaimed more of Kowloon Bay to expand the Kai Tak airfield. All three parts of the Kowloon City Pier were buried under this reclamation, and disappeared from sight.
2000s - Re-discovery
Kai Tak airport was closed in 1998, but it wasn't until 2008 that excavations revealed that at least some of the old pier had survived. Further excavations showed there were significant remains. It was decided to re-bury and preserve the remains, with the plan to later excavate them again and make them accessible to the public.
As I write this in August 2022, that excavation work is currently under way.
- See the numbered item in Annex B of the AAB's Board Paper ref: AAB/30/2009-10, "Remnants of Lung Tsun Stone Bridge in Kai Tak Area".
The board paper gives a brief overview, with more information in the four annexes:
- Full text of "Convention between the United Kingdom and China Respecting an Extension of Hong Kong Territory".
- Item 29 of the Public Works Department's Annual Report for 1899, and Item 14 in their report for 1900.
- 1903 Map of Kowloon
- Item 37 of the Public Works Department's Annual Report for 1906.
- Item 86 of the Public Works Department's Annual Report for 1908.
- Item 89 of the Public Works Department's Annual Report for 1909.
- Item 93 of the Public Works Department's Annual Report for 1910.
- Kai Tak Bund reclamation
- Tenders invited for Raising Level of Causeway to Kowloon City Pier: GA 1934 (suppl) no.97
- Item 184 of the Public Works Department's Annual Report for 1934.
- Tenders invited for Reconstruction of Kowloon City Ferry Pier: GA 1936 (suppl) no.273
- Item 178 of the Public Works Department's Annual Report for 1937.