The context is clear in the previous paragraph. Charles George Gordon was referring to the end of "old arms and tactics of Frederick and Napoleon". Here's the full context:
It was in the battle of the Tchernaya, fought in August 1855, that the first foundations of the present kingdom of Italy were laid; and while the arms of France, England, and Russia were proving at Alma, Inkermann, Balaklava, and in the ten weary months of toil and bloodshed of the trenches, that the old arms and tactics of Frederick and Napoleon had become wholly obsolete, Moltke was quietly watching the costly experiment, and had already begun to apply to the Prussian army the lessons of change and improvement which the errors and failures of the rival antagonists were teaching.
"The last of the old sieges," Gordon had called that of Sebastopol. Of the old battles, old arms, old cannon, he might have added, and sorrowfully also have said, "the last of the old army too." ...
Charles George Gordon by Lt.-General Sir William F. Butler
So he's probably referring to the siege as the last using those old arms and tactics. The Crimean War as a whole marked many turning points in military technology and tactics - the Charge of the Light Brigade being a famous example, arguably the last of its kind. The changes were brought about by advancements in firearms technology, which Moltke the Elder observed. These changes continued to be seen in the Civil War, where we saw repeating rifles. The devastating effect of newer cannons and firearms made it more effective and crucial to perform coordinated, enveloping attacks, and reduced the effectiveness of line-breaking charges.