The first police station of which I'm aware was at the western end of High St, near the Whalers Tunnel under Arthur Head, and photographs showing it are on the page for the WF Samson house, as it was next door. I should think the officers used the Round House as their lockup when required, as it was just up the stairs to the west.
The Henderson St Police Station complex was built between 1896 and 1903, and includes the former courthouse, police station, various accommodation buildings, artillery drill hall, and lock-up. According to Conole (2006), the police only moved here in 1916 from High St. (They are currently back in High St, in a former bank building.)
Skip Watkins 1985 photo FHS #E000254. Caption: The former Courthouse, constructed of Fremantle limestone in 1899, is an example of a building designed in the Federation Academic Classical style by the Public Works Department. Fremantle Courthouse (fmr) and Police Station Complex formed part of the original convict land grant that was established as part of the Fremantle Convict Establishment in 1851. The site, originally used for barracks and warders’ quarters, was taken over by the police in 1888, and since then has maintained links to both the Police and Fremantle Prison.
Skip Watkins 1985 photo FHS #E000252.
The first substantial buildings on the Henderson St site, where the police station complex is now, were barracks built for the sappers (later Royal Engineers) who began to arrive in 1850 to construct and manage the infrastructure required for the convict enterprise, the largest group arriving in 1851 on board the Anna Robertson. The image above is clipped from an 1885 map and shows the sappers barracks at the top, next to the ST in HENDERSTON ST. The long building was the married quarters, the bottom one was for single men, and the top building housed the offices. (Kerr 1998: 2) At the other end of Henderson st, on the corner of South Terrace, are two pairs of semi-detached houses which Kerr shows as being for 'instructing warders'. These, as opposed to 'discipline warders', who lived in the Warders Cottages between, were sappers whose job it was to instruct convicts and supervise their work in building etc. It's not clear which of the sappers got these separate cottages. The building at the bottom of the image, with '28', was called The Barracks, and housed members of the Enrolled Pensioner Force.
Murray Edmonds 1971 slide photo FHC #ES00303, with this caption: Police station and courthouse (1898-1899) and Warders Quarters in Henderson Street looking towards South Terrace. Three sets of Warders Quarters are shown from left: Nos 41-33 (built 1852), Nos 29-19 (built 1851) and Nos 17-7 (built 1858).
Part of the former Fremantle police complex with the entrance to the courthouse on the left. Taken from the Henderson St carpark by Derek Graham for Google.
See also: courthouses.
Conole, Peter 2006, 'Fremantle 1919: A Slice of Policing Life', Western Australia Police Historical Society.
Conole, Peter 2010, 'Policing the port in early colonial times', Fremantle Studies, 6: 12-28.
Wikipedia page for the Fremantle Police Station complex
Heritage Council page for the same (but the link is broken)
Garry Gillard | New: 13 May, 2017 | Now: 14 May, 2017