The Anchor was run by a John Sandford and was in Castle Square in 1823 according to Pigot. Is listed as being in Castle Street by Besley (1828). The Castle Inn and this place may well be the same place.
The EFP printed a "for sale" notice for a Anchor Inn in Castle Lane as early as 1773, Thomas Bull was landlord.
The Castle Hotel at 20 Castle Street is first listed by Kelly in 1889. Again, the Anchor may have become the Castle Hotel. The latest listing I can find for the Anchor is 1859.