Moshe Dayan, the famous general with the eyepatch who had taken part in all of Israel's shooting wars, arrives here today as a combatant in a battle for American public opinion.

It is a different kind of struggle, in some respects unprecedented in the 30-year history of the U.S.-Israeli alliance. For all the ups an downs of official state-to-state relations, Israel has had a clear field in the minds of Americans. But since Egyptian President Anwar Sadat's dramatic journey to Jerusalem in November, an Arab competitor for the first time is seriously challenging Israel for the favor and support of the U.S. public.

President Carter's decision to sell jet airplanes to Egypt as well as Israel is the most tangible symbol of a continuing Washington shift toward equal treatment for the two Middle East antagonists. The arms decision was made politically possible by a sharp rise in pro-Egyptian sentient in this country.

Public discomfort about providing weapons to both sides may undermine the very trends which helped to bring it about.

As foreign minister of Israel's Likud government, Dayan has been the point man of his country's rebuttal to Sadat since the Egyptian leader left the United States - and Dayan arrived - early last week.

On a "Meet the Press" interview and fast-paced appearances in New York, Chicago, Los Angeles and Chicago, Dayan sought to place the blame on Sadat for slow progress in the peace negotiations, and argued against American supplies of weapons to Egypt and Saudi Arabia.

In travels sponsored by the United Jewish Appeal and Israel Bonds, which are trying to raise $700 million for Israel this year, Dayan has stiffened the political as well as material support of the American Jewish community.

In Chicago, he told a news conference that he had detected no lessening of pro-Israeli setiment among American Jews, adding that "realistically, people here and in Israel were very impressed by Sadat. After all, he's the first Arab leader who was willing to come to us and talk peace."

Rep. Abner J. Mikva (D-Ill.), who was present at private Chicago talks with Dayan, said American Jews realized that "at this time Sadat has the momentum and is controlling what will be discussed and when."

Los Angeles political fund-raiser Mickey Kantor, after sessions with Dayan there, reported concern and anxiety among American Jews about the public at large, but said that the Jewish community is standing firm, especially after the announcement of proposed arms sales to Egypt and Saudi Arabia.

In pronouncements in Los Angeles to Jewish groups just after the arms announcement, Dayan seemed "more moderate than he needed to be," according to Kantor, who said he felt the Israeli official was trying to keep the Jewish community from "over-reacting to the Carter administration."

At a glittering candlelight dinner in downtown Los Angeles Tuesday night, with cameras barred and only four reporters listening in from a separate room, Dayan loosed his emotions in discussing the negotiations with the Arabs.

According to Washington Post correspondent Lou Cannon, Dayan received his loudest applause when he vowed that Israel would not give up the strategic Sinai port of Sharm el-Sheikh, and again when he declared, "Don't worry, we are not going to pull out from the West Bank."

Today Dayan is to state his case officially to Carter, Secretary of State Cyrus R. Vance and other high officials at the White House and State Department, but a closed-door session with the House international Relations Committee and a kleig-lighted news conference may be equally if not more important.

As announced yesterday in Jerusalem and Washington, Prime Minister Menachem Begin will take up the Israeli case with both officialdom and the American public in a trip here in early March.

In the meantime, Moshe Arens, the American - educated chairman of the Israeli parliament's foreign affairs and defense committee, continues his 16-day tour of the United States to counteract the impact of Sadat. Another Israeli parliamentarian has just been here on a similar mission, and still another is expected next week.

"There is a struggle going on here," said an Israeli official yesterday. In the wake of Sadat's visit, "It's not enough for us to send written statements around. We have to send our senior people."

Prof. William Schneider of Harvard's Center for International Affairs, co-author of a recent extensive study of American opinion toward the Middle East, said in a telephone interview yesterday that "Israelis are concerned that they are losing the public relations war with Egypt in the United States."

Schneider said there is no question that "they are losing it in a relative sense, because Egypt has so improved its image." But he added that there is yet no clear evidence that public sympathy for Israel has declined as sympathy for Egypt has sharply risen.