Born in Kaufbeuren, West Germany, Sulzer went through the youth system of the localESV Kaufbeuren. He first received ice time with the men's team in the third-tier Oberliga in 2000–01 as a sixteen-year-old. Two years later, in 2002–03, he split time between ESVK's team in the second-tier 2. Bundesliga and the Hamburg Freezers of the premiere Deutsche Eishockey Liga under Förderlizenz. In the subsequent off-season, Sulzer was selected by the Nashville Predators in the third round of the 2003 NHL Entry Draft, 92nd overall. Following his draft, Sulzer remained in the higher-profile DEL, transferring to the DEG Metro Stars. He spent four seasons with the Metro Stars, recording a DEL career-high 18 points over 48 games in 2005–06. He also helped the team reach the playoff finals that year, recording 9 points over 13 post-season games. in 2013 On June 1, 2007, he signed a two-year contract with the Predators. Moving to North America, to begin playing within the Predators organization, he was assigned to the team's American Hockey League affiliate, the Milwaukee Admirals, for the 2007–08 season. He recorded 32 points over 61 games as a rookie in the AHL. The following season, he improved to 34 points over 48 games with Milwaukee, while also earning his first NHL call-up, appearing in two games for the Predators. Sulzer became a more regular addition to the Predators' lineup in 2009–10, dressing for 20 NHL games, while also recording 30 points over 36 games in between call-ups. Sulzer scored his first NHL goal on January 23, 2011 against Devan Dubnyk of the Edmonton Oilers. The following month, on February 25, he was traded to the Florida Panthers in exchange for a conditional draft pick. Splitting the season between the Predators and Panthers, he recorded 5 points over 40 games. During the off-season, Sulzer became an unrestricted free agent. On July 7, 2011, he signed with the Vancouver Canucks. He made the team out of training camp, but was a healthy scratch for the first three weeks of the season before dressing for his first game as a Canuck on October 26, 2011. On NHL trade deadline day, Sulzer was dealt to the Buffalo Sabres, along with forward Cody Hodgson, in exchange for defenseman Marc-Andre Gragnani and forward Zack Kassian. On May 21, 2012, the Buffalo Sabres re-signed Sulzer to a one-year contract extension worth $725,000. In explaining the extension to media gathered at First Niagara Center, GM Darcy Regier said, " played very well with Christian. And I think the other thing you really notice out there is his composure; his ability to not only break up plays defensively, but to make plays offensively with the patience that he showed. We were pleasantly surprised with his play, and that moved us toward getting him signed." For the 2013-14 season, Sulzer spent 25 games with the Sabres and 10 games with the Rochester Americans of the American Hockey League. He sustained a concussion on March 13 and missed the remainder of the season. Sulzer signed a five-year contract with Kölner Haie on May 1, 2014. At the conclusion of his contract in Cologne, Sulzer opted to leave as a free agent and return to former club, Düsseldorfer EG, then known as the Metro Stars, signing a one-year contract on May 9, 2019.
