Robson, a recent £600,000 purchase from Celtic, was dismissed in the 56th minute when shown his second yellow card for a foul.
Middlesbrough hoped to follow up their impressive 4-1 midweek win over Doncaster against a side reeling from a 6-0 rout by Cardiff City.
In his programme notes, Boro manager Gordon Strachan wrote: "I'm not going to mislead the fans by wearing a scarf and kidding them I have been a Middlesbrough supporter since I was a youngster, but I will strive to give our supporters a team they are proud of. It might take a few months, or it might take a year, but that is our aim.
"Our form in a lot of matches has been good but we have not been getting the results that our performances deserve."
There was pre-match speculation as to whether Boro's England Under-21 international winger Adam Johnson would be making his last appearance at the Riverside Stadium.
The highly-rated Johnson has been linked with Manchester City and Boro fans will await to see whether the super-rich club increase their reported £5million bid before Monday's transfer deadline.
After a quiet opening, Willo Flood had a shot blocked by the City defence after a brilliant run.
Middlesbrough enjoyed the greater possession but failed to achieve a breakthrough against a packed defence.
Robson was booked for a foul after 32 minutes.
There was a lack of atmosphere in the stadium and the game desperately needed a goal to lift the crowd.
Jamal Campbell-Ryce was lively for the visitors but he lacked support.
Immediately after the break, Paul Hartley blasted a shot wide for the Robins.
For Middlesbrough, Johnson broke down the right, cut inside and then unleashed a fierce shot wide.
Chris Killen appeared to miss a glorious chance for Middlesbrough but a linesman's flag had been raised.
Campbell-Ryce tried a long distance shot from near the touchline that had Danny Coyne back-pedalling to check the danger.
In stoppage time, the unmarked David Wheater missed a glorious scoring chance for Middlesbrough when he headed wide from a Gary O'Neil corner.
Bristol City manager Gary Johnson will have been pleased with the resilience of his side after the Cardiff rout, but overall this was a poor advertisement for the Championship with limited thrills and entertainment value.