A vaccine requires time, often days if not a few weeks, for it to be absorbed by the immune system and the immune system to produce a sufficient level of antibodies to provide immunity. Giving the Russian officer the vaccine and then immediately exposing him would not work.
When Dr. Scott arrives on the Russian cruiser she is patted down and searched by a Russian officer wearing the rank insignia of a Lieutenant. The officer he orders to search the case the virus samples are in is wearing the insignia of a Senior Lieutenant, meaning he outranks the other officer yet seems to be taking orders from him for some reason.