The line that Spock speaks during the movie that made absolutely no sense to me was something like "Thrusters on full" during Kirk's promotion scene. I kept expecting to cut to the Enterprise and have Kirk use that exact line (kinda as a tie in to between the two timelines). However, since it wasn't quoted exactly I have no clue why Spock said it.
Exactly as you said. More than once in the movies before moving out, Kirk gives that command to Sulu. When Spock says it upon watching Kirk finally take his rightful command, we then cut to Sulus typical thruster/impulse lines and a one by one to each of the crews customary moving out lines per the movies.
Basically Spock was saying to himself what he knew would come next given how often he remembers it happening on the Enterprise.
Its kinda of a old man remembering his past scene, and was really meant for the fans than it was the general audience.