Two or three effects arise from Gatsby's use of the words, "old sport," in The Great Gatsby."
In addition to Gatsby possibly attempting to appear British or high class, Gatsby also appears to use the phrase to endear himself to others. It is a phrase that Gatsby uses to show acceptance of others. It shows he accepts others.
At the same time, however, the words, "old sport," probably reflect Gatsby's inneptness at social contact. The words can be a bit annoying. He uses it for people that consider themselves his friends, but he also uses it for people who do not consider themselves his friend--most notably, Tom, who throws it back in his face. It seems to be something Gatsby says that is awkward and overused, and he uses it with mixed success.
Finally, the words suggest a lack of sincerity on Gatsby's part. And this perception of the words is accurate. When it comes to anyone accept Daisy, and possibly Nick, Gatsby is insincere in his friendship. He is obsessed and consumed by his illusion of Daisy and of their relationship. And he cares little for anything else.