There are 38 steps from the Goth's starting space to Rome. If you never give tribute the entire game, the Goth will reach Rome in 38 player turns. If you give tribute every player turn, the Goth will reach Rome in 39 players turns (37 tributes + Goth move + Goth move). Unless my calculations are incorrect, the most you can delay the Goth is 75 turns. This is accomplished by alternating paying tribute and not paying tribute. Every two player turns, the Goth will move 1 space.
Every time you do what the previous player did, the Goth gains a space (real or virtual). Every time you do the opposite of the previous player, the Goth is delayed. So it seems that if you have a significant lead, you should just "follow suit" as much as possible. And if you are behind, you should not follow suit if possible. Of course, the specifics of which resources you have, need, and what cards will be destroyed overides all this.