As a player, Rubčić played for HNK Rijeka for much of his career, collecting 184 league caps between 1982 and 1996. He was club's captain for 5 years, and he was also part of the first Croatian national football team after independence, in 1990. He debuted on 22 December 1990, in a friendly match against Romania, in Rijeka. He came on as a substitute in 86th minute instead of Zlatko Kranjčar. The result was 2-0 for Croatia.
He prefers fast playing with lot of running during the whole match, emphasising on the technical aspects. He said that the national team will from now on play 4-2-3-1 formation, which fits the team best. In the 2010 Asian Games, people had expected a lot from Robert but he couldn't stand up to the expectations of the people as Bangladesh crashed out of the Asian games, losing every game. He was sacked by Bangladesh Federation on 14 June 2011 so close to 2014 World Cup qualifier against Pakistan. He had already left Bangladesh on June 2.