George Bush, a virtual unkown in national politics a few months ago, established himself tonight as a serious challenger to Ronald Reagan by running neck-and-neck with the former California governor in the Iowa precinct caucuses.

With about 43 percent of the state's 2,531 precincts reporting, Bush had 33 percent of the vote. Regan, who was criticized for campaigning so little in the state, had 29 percent, CBS and NBC were projecting a virtual dead heat by the time the counting was completed.

Senate Minority Leader Howard H. Baker of Tennessee was a distant third with 14 percent, while former Texas governor John B. Connally, who had hoped for a third-place finish, was fourth with 9 percent. Rep. Philip Crane of Illinois was running close to Connally in fifth place, while Rep. John B. Anderson of Illinois, who campaigned only briefly here, was leading Kansas Sen. Bob Dole in the race to avoid to cellar. Dole may now be expected to drop out of the presidential race and seek relection to the Senate.

The straw ballot reported by the Republicans did not relate directly to delegate strenght. It was simply a beauty contest taken as Republicans entered neighborhood caucuses. The selection of delegates was done on a separate ballot.

Regardless of the eventual winner, Bush accomplished what he had hoped for, which was to come close to Reagan and separate himself from the other GOP candidates and go into New Hampshire with new momentum. Baker stayed alive in the race, while Connally goes to the next stage with a newly damaged campaign.

Bush's strong showing was the result of a superior organization and the personal time he had spent stumping the state. A CBS poll taken last weekend found Reagan more popular than Bush, by 24 percent to 14 percent.

Bush came before a cheering crowd f supporters and said, "What's wrong with good organization and working harder than the next guy? Because of what's happened in Iowa we are going to go all the way to the White House."

Connally said he was "never satisfied" not to win but refused to rule out hope he might finish third in front of Baker.

CBS News estimated that more than 100,000 Republicans turned out for the caucuses, more than four times as many as in 1976.

The results were seen as a severe blow to Reagan and the strategy of his campaign chairman John Sears, who kept Reagan out of the Republican debate here on Jan. 5 and had him campaigning all last week in the Northeast even as a state poll showed Reagan's strength slipping.

Sears said he felt Bush had "separated himself from the pack as far as this event is concerned," but expressed no concern for what the results might do to Reagan's prospects.

Baker said, "I predicted that we would compete for third place and that, while you'd like to be second or first, we accomplished what we set out to do. In the span of a relatively few days, we put together a campaign organization that moved from nowhere to somewhere. The campaign for the Republican presidential nomination has just begun."

Regan was the early favorite in the Iowa caucuses, and until recent weeks he had been expected to win handily.

His roots here are deep. He began his carer as a sportscaster on a Des Moines radio station in the early 1930s. In 1976, he won 49 percent of the Iowa delegates to the Republican National Convention n his race against President Ford.

As recently as December, the Des Moines Register's Iowa poll found that Reagan had the support of 50 percent of the state's Republicans. His standing began dropping dramatically when he refused to appear in a nationally televised debate with six major GOP opponents on Jan. 5. More than one-third of the state's Republicans watched that event, and when a new Iowa poll appeared on Jan. 11, Reagan's popularity had dropped to 26 percent.

Virtually unknown a year ago, Bush used the same strategy for this year's Republican caucuses that Carter used in the 1976 Iowa Democratic caucuses, which propelled him from political obscurity. Bush campaigned in the state early and often, logging 27 days here since March.

With early help from Iowa's national committeewoman, Mary Louise Smith, he recruited a growing number of the party's moderates. Before his opponents knew what had happened, much of the party's structure here had signed with him. By mid-December, one of every five GOP voters in the state had been contacted by mail or personally.

With only 37 delegates, Iowa will nob be a major factor at the Repubican national convention But the candidates and the media have turned the precinct caucuses into the earliest political spectacle of the 1980 presidential campaign -- in effect preempting the New Hampshire primary as the first important test of the year.

The caucuses are small neighborhood meetings, held in each of the state's 2,531 precincts. On the Republican side, 44 percent of them draw so few people -- fewer than 7 percent of the registered partly members attended in 1976 -- that they can be held in living rooms.

The GOP caucuses are a two-tiered affair. The tier candidates watched tonight was the straw ballot "beauty contest" that was taken as voters entered the caucuses.

These beauty contest results have only limited value in predicting the number of delegates each candidate will capture eventually for the national convention.

Delegates were to be picked in a long, complex process that winds through precint, county, congressional district and state party conventions ending on June 7. Tonight each precint selected delegates to the state's 99 county conventions in voting entirely separate from the beauty contests. The delegates were not required to indicate which presidential candidate they support. Traditionally Republicans picked for these spots are local party favorites, not legally bound to support any candidate.

Although the Democratic race has dominated attention here, the Republican battle has stirred more interest in the party caucuses than any contest since 1952, when Gen. Dwight Eisenhower defeated Sen. Robert A. Taft in a caucus battle that attracted 100,000 voters.

In a sense, the results were being watched more closley for indications of poor showings than of good. It was a test of vastly different strategies.

Reagan was on the spot as the early front-runner who refused to participate in a debate with his opponents and made only perfunctory appearances here. Bush, who showed up only as an asterik in polls here a few months ago, attracted early media attention by building up one of the most extensive campaign organizations the state has ever seen. This raised expectations for a good showing.

His supporters banked on a lowturnout, hoping that the party regulars he had wooed would dominate the caucuses. The other candidates all hoped for a large turnout.

After finding his campaign in disarray in early November, Baker counted on a rescue mission performed by popular Gov. Robert Ray and an extensive media campaign to keep his candidacy alive. Connally, 62 also used an extensive television advertising effort, and crisscrossed the state last week end in a 40-hour marathon to demonstrate he is more vigorous than Regan, who will be 69 next month.