Think I finally got it actually! Posting below in case anyone else wants to know! (Or in case I goofed and someone wants to correct me lol)
public void RequestRank(string userName)
{
var query = LeaderboardSearchQuery.ByUsername(userName);
LeaderboardCreator.GetLeaderboard(publicLeaderboardKey, false, query, (rankData) =>
{
rank = rankData.Rank;
});
}