I think it was in the Waterloo Street area, but can't find it so far on maps from 1874 and 1911, it definately existed (somewhere!) in 1851 and there was some clearance there about that time.
I suspect it was wiped out by the railway junction between Jackson Streey and Old Chester Road (now Tunnel Road). Also missing is Meacock Street and Tory Place. Alternatively there is a block of housing between what is now Tunnel Road and Chester Street, I can't find detail for that area.
There may also have been a Rushton Court in Liverpool by Prussia Street (off Pall Mall) as well.
I don't know of any street maps for around 1850s Birkenhead, I guess the archives may have them, perhaps Birkenhead Central Library.