The 18-year-old has made a quick impact at the Santiago Bernabeu since arriving from Santos in a reported €45million deal agreed last year, scoring five goals in six appearances.
Rodrygo has already earned comparisons to Ronaldo, who scored a club-record 450 goals for Madrid before leaving for Juventus in July 2018.
However, the teenage forward – in line to earn his first senior Brazil cap in Friday's friendly with Argentina – is cautious about playing with such high expectations.
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"Being able to play alongside Cristiano Ronaldo at Real Madrid was a dream of mine and it's a shame that he left because now I think it would be very difficult to play with him," he said at a Brazil news conference on Tuesday.
"I am still a very young player to talk about filling the place left by someone like Cristiano Ronaldo, who is one of the greatest players in the history of the club – if not the greatest.
"I don't want to weigh myself down with the pressure of replacing Cristiano Ronaldo."
The youngster has made a big impression in the Spanish capital but is surprised by how quickly his career has taken off.
"You always try to make sure you are fully prepared so that things turn out well, but I could never have imagined that everything would happen so quickly," he said.
"It's all come as a huge surprise to me. Only a short while ago I was hoping to make my Champions League debut... and maybe at the most score a goal.
"Scoring a hat-trick like I did was something nobody expected. So I can't really explain how I felt after the game. Al I could do was enjoy a special, once-in-a-lifetime moment."
He added: "Learning the ropes as a footballer isn't just about what's done on the training pitch. There is a big mental part too, preparing yourself so that success doesn't go to your head and knowing how to handle fame and to not be presumptuous.
"On that side of things, it helped me a lot that my father was a footballer, and that gave me a head start on the mental part. I watched him go through things that are happening to me now and so I know I can't let all of this attention go to my head."
Rodrygo was left on the bench for Madrid's 4-0 win over Eibar on Saturday but is primed to make his senior Brazil debut.
"I feel fulfilled. It has always been my dream," he said.
"Since I was a boy I have dreamt about representing the national team and playing alongside the best players in the world – players whose careers I have been following for a long time.
"So being here gives me happiness that I can't begin to put into words. Everything that has been happening in my life makes me feel very happy."