The times we are given some good distance / time information in TOS was "Obsession" and "That Which Survives" and that pretty much tells us that going between star systems the Enterprise can easily do 1,000 LY in a day. TOS speeds were generally faster when it came to interstellar travel so going to the galactic edge wouldn't have taken too long.
Allow me to add this piece of information from "Bread and Circuses":
KIRK: Mister Spock, assuming that the wreckage drifted at the same speed and direction for the past six years?
SPOCK: It would have come from planet four, star system eight nine two, directly ahead.
CHEKOV: Only one sixteenth parsec away, Captain. We should be there in seconds.