The text does not tell us clearly how many days Rainsford was out in the island. However, if we follow the storyline and make an educated guess, he spent two nights on the island.  The first night he spent on a tree, after concealing his tracks.  It seems that Rainsford's first plan was to hide.  When Rainsford realized that Zaroff easily found him, he decided to make traps, which killed Ivan and even wounded Zaroff.  

When this happened, Zaroff went back home.  The next day Zaroff was back.  We do not know where Rainsford slept that night, perhaps he did not.  In any case, it was another night. 

This is when Rainsford had an idea.  Instead of trying to hold out for three days, he needed to go on the offensive.  So, he swam to Zaroff's house and waited for him there.  Here is what Rainsford said to himself:

Rainsford knew he could do one of two things. He could stay where he was and wait. That was suicide. He could flee. That was postponing the inevitable. For a moment he stood there, thinking. An idea that held a wild chance came to him, and, tightening his belt, he headed away from the swamp.

To win "The Most Dangerous Game," how many days must Rainsford survive?

General Zaroff spells out the specifics of his "most dangerous game" after informing Rainsford that he will be the human prey on Zaroff's next hunt. Zaroff gives Rainsford several options: He can play the game or else die at the hands of Ivan. Zaroff will give Rainsford a day's head start; the Cossack will then begin to track him at dusk, since 

"Hunting at night is much more exciting than by day, don't you think?"

Zaroff will give Rainsford three days. If Zaroff has not killed him by that time, he will declare Rainsford the winner. Rainsford takes the entire three days to elude Zaroff before implementing his unusual twist on the game.