As already observed, Obama did run for the US House in 2000.
https://www.ourcampaigns.com/RaceDetail.html?RaceID=141037 I'd have to say that he probably never had much of a chance against Bobby Rush, though.
What sealed Obama's doom was no doubt missing a gun control vote in Springfield while on an extended Hawaiian vacation. (Obama later explained that he extended it because Malia, then 18 months old, became ill.) Most of the voters in the First Congressional District were probably unaware of Obama's Hawaiian roots, and the idea of escaping a Chicago winter by going to Hawaii is just a dream to most South Siders. It all played in with Rush's "plain folks like you and me versus some professor from Hyde Park" theme. But even without Obama's missed vote, Rush would probably have won. Obama just was not well known enough outside his own state senatorial district.
BTW, if somehow Obama had won the House seat in 2000, it is by no means clear he would have run for the Senate in 2004. He would have a safe congressional seat (at least in the general election, and there is no reason he would be particularly vulnerable in primaries) and might not want to give it up for what looked at the time like a long-shot Senate race. (By contrast, he had nothing to lose in OTL since his State Senate seat did not come up for re-election until 2006.)
Very likely Obama would stay in the House and be typecast as a "black Congressman" representing a "black seat" (IL-01 was the first northern congressional district to elect an African American congressman, something it has done continuously since 1928) in a way he was not in OTL.