The Ultimates imprint is separate from Marvel's bigger universe. The line debuted in 2000 with “Ultimate Spider-Man.” The “Ultimate” version of Captain America debuted in “Ultimates” No. 1 by Mark Millar and Bryan Hitch in 2002.
The Marvel Universe Captain America faced the possibility of running for president back in issue No. 250 of “Captain America,” in 1980, but Captain America ultimately declined the nomination.
A follow-up story, featuring what might have happened if Captain America had won the election, was published in “What If” Vol. 1 No. 26.