The Armstrong was not mentioned in dialogue. Its registry, confirmed by the Blu-Ray special "Starships", was seen on a starship in front of the Enterprise before the fleet warps to Vulcan. The ship's appearance was identified in Star Trek - The Art of the Film.
A successor to this ship was in operation by 2261, when it visited Deep Space Starbase K-11. Yuki Sulu, a sister of Hikaru Sulu, had worked in that ship's engineering department before her transfer to the Enterprise. ("The Khitomer Conflict, Part 1")