Adrian Pye tells us

The Victoria Arms and Blacksmith's Arms is the same pub.
It seems that while William Warren was the landlord the name Blacksmith's Arms was used unofficially but all licenses issued were in the name of the Victoria Arms. William Warren died in 1899 and thereafter it was again called the Victoria Arms.
In 1866 the Victoria Arms was known to be a pub but there are no earlier records. Closure was around 1907.

The map location is based on what's now 23 Ingate, on the assumption that the numbering scheme is unchanged. The building now at number 23 cannot be the original pub building as it sits back from the line of houses, which was not the case on the OS map of the end of the 19th century.

It evidently closed between June 18th and 24th 1907 as on the former date it's listed as for sale as the Victoria Arms, but on the latter as "formerly known as the Victoria".