Ending what has been an arm's length relationship with the Supreme Court, President Reagan will host the justices at a White House luncheon Oct. 1, three days before the court term opens.

Once it was customary for the justices to visit the president on the first Monday in October, a tradition that Reagan would like to restore. Chief Justice Warren E. Burger apparently would like to restore it, too, and he explored the matter Aug. 6 at a luncheon with White House counsel Fred Fielding. The result was a formal invitation to the White House for the justices, and they accepted.

Burger, a former Republican politician, shouldn't find it difficult to establish a relationship with Reagan, who is prouder than ever about the precedent, the politics and the results of his appointment to the court last year of Sandra Day O'Connor.

To those who worry that presidential hobnobbing with Supreme Court justices infringes on the separation of powers, administration officials are likely to point out that this has been happening since George Washington consulted the first chief justice, John Jay, on the content of the 1790 State of the Union address. Reagan has made no secret that he would like to establish a closer relationship with Burger, whom he barely knows.

The White House also has a keen interest in the retirement plans of justices, a delicate subject unlikely to come up at the luncheon. While rumors persist that one or more justices will step down after the present term, White House officials say they know of no imminent retirement.

White House press secretary James S. Brady and his wife, Sarah, will take their first vacation this week since Brady was severely wounded in the assassination attempt March 30, 1981. The Bradys will spend a week in Chicago where, among other things, Brady will throw out the first ball at a Cubs game Sept. 20. It will be "Jim Brady Day" at Wrigley Field, honoring one of the most devoted Cubs fans of all time.

Brady continues to make progress. Four weeks ago, he walked unaided, but he then fractured some small bones in his back because a blood-thinning medicine had made them brittle. Doctors have prescribed a change in medication, and Brady is expected to be walking again in a few weeks.

"It's uphill," Sarah Brady says of their long struggle. "There are some temporary setbacks, but in the long run he always keeps going uphill."

Public support for Reagan increased four percentage points in a survey taken by GOP pollster Richard Wirthlin after congressional approval of the $98.3 billion tax bill. The poll predates his Middle East initiative and the override of his supplemental appropriations veto . . . Several Republicans, including some in the White House, think the veto would not have been overridden if the president had stayed here rather than campaigning in Utah and sending Republican Sen. Orrin G. Hatch back to vote . . . Reagan will campaign in New Jersey Friday, in Ohio Sept. 21 and in Virginia Sept. 29.

The word from the White House is not to expect further initiatives from the president during the election campaign. Reagan will work from his basic domestic theme of portraying the Congress as full of irresponsible "big spenders" determined to undo the budget-cutting president. He'll also get all the mileage possible from the Middle East initiative, which helps undercut lingering impressions of Reagan bellicosity that hurt him with women voters.

The only exception to this, and it isn't much of an exception, is the crime message that Reagan is expected to send to Congress today. Most of it is a restatement of previous positions and support for pending legislation.

White House strategists are convinced that former Secretary of State Alexander M. Haig Jr. will run for president in 1984 . . . Despite widespread reports, in this column and many other places, of a post-election White House shakeup, the betting is that chief of staff James A. Baker III, deputy chief of staff Michael K. Deaver and White House counselor Edwin Meese will still be around. That assumption is based on the belief that Attorney General William French Smith won't decide to quit and leave a vacancy for Meese . . . There are likely to be some Cabinet changes, though. The White House still wants Secretary of Transportation Drew Lewis to replace Richard Richards as GOP chairman, even though Lewis is cool to the proposal. And if Secretary of Labor Raymond J. Donovan leaves, some insiders think his replacement could be special trade representative William E. Brock.

A mild stir was caused in the White House last week by Gary F. Schuster's report in The Detroit News that Reagan had written off unemployment-ridden Michigan and would not campaign there for GOP gubernatorial nominee Richard Headlee or Senate nominee Philip Ruppe. Administration officials tried to mollify the offended candidates but confirmed that the president had no plans to visit Michigan.

Reagan will, however, spend three more days at his California ranch in October, sandwiched between campaign appearances in Texas, New Mexico and Nevada.