As direct negotiations with Israeli open here this week, President Anwar Sadat rides higher than any Egyptian leader since the time of pharoah. "He has hit a gusher of public support," one American diplomat said, "that even he didn't know existed."

He has asserted with a vengeance Egypt's primacy in the Arab world. He has opened - against the will of Moscow and without the support of Washington - a road to settlement of the world's most dangerous conflict.

Sadat's new popularity is not among the sophisticated political classes. Left-wingers bear him a grudge for his anti-Soviet stand and former Nasserites condemn his refusal to play off Russia against America. Fanatic Moslems oppose his efforts to modernize Egypt, not to mention the luxurious tastes of his wife.

That opposition is balanced by support from the right, especially the army, which is Sadat's power base. Senior military officers in particular like the opening to Israel, partly because it expresses an Egypt-first policy, partly because it gets them off the hook in the wake of the 1973 war - which is to say while they are even, if not ahead.

The new element in public opinion is the mobilization of the unorganized masses. "Sadat has been plebiscited," one Egyptian friend said to me after the enormous popular demonstrations in favor of the president on Thursday.

An official of the Interior Ministry, asked whether he was concerned about the security of the Israeli diplomats and journalists coming here for the negotiations, replied: "No, I only hope the people don't applaud them too much."

Enthusiastic domestic support has enabled Sadat to bring to bear upon the Arab world Egypt's geographically central position and enormous superiority in population and military power. he went to Jerusalem largely because he became convinced that efforts led by Syria to organize a common Arab position for the Geneva conference would yield a set of proposals sure to be rejected by Israel.

Arab reaction to the visit ranged initially from hostile to reserved. Sadat has played hardball with those who were hostile. When Syria, Iraq, Algeria and Libya announced at the conference of radical Arabs last week that they would "freeze" relations with Egypt, Sadat one-upped them by breaking diplomatic relations. He tells visitors that President Hafez Assad of Syria suffers from a "complex of impotence." He says that Palestinian leader Yasser Arafat is "weak."

As to Arab reservations, King Khalid of Saudi Arabia did not like Sadat's going to pray in Jerusalem at a time whe n the form called for a pilgrimage to Mecca. King Hussein did not want to move without Saudi support - particularly if Syria opposed it. But Sadat has apparently eased King Khalid's misgivings, and all signs are that KIng Hussein is planning to enter the Cairo negotiations sometime.

As to the superpowers, Sadat's peace initiative was, in part, directed against them. He saw the Carter administration, despite brave talk, unable to wring concessions from Israel. He also saw Washington focus on bringing Syria and the Palestinians to a Geneva conference as if they were more important than Egypt. He was particularly enraged when Washington, in a joint statement on Geneva with Moscow, handed the role of bringing Syria to the conference to his archenemies, the Russians.

So, after slamming the Syrians by breaking relations, he invented a story that Moscow had been pushing Damascus to oppose his peace initiative. On that pretext he then slipped another wet mitten to Russia by closing down the Soviet bloc's consulates and cultural centers in Egypt.

Toward the United States Sadat remains friendly - but far more independent than before. He referred to one of Washington's working papers for the Geneva conference as "insulting." His trip to Jerusalem followed repeated American rejections of several projects suggested by Sadat as substitutes for Geneva.

He invited the Israeli negotiators to Cairo by a direct approach at the United Nations - not through the American channel. He tells visitors that one of his present aims is to "take the pressure of the Jewish lobby off Carter" - a distinctly patronizing bit of assistance.

Despite this show of strength, to be sure, Sadat is not yet out of the woods. He has talked publicly of having to resign, and while that is largely to stimulate demands that he stay, he could in fact fall on his face. Everything now depends on whether the Israelis in the talks getting under way here (and no doubt elsewhere) are prepared to make Sadat the generous offer, perticularly with respect to the occupied territories West of the Jordan River, that he needs to go on serving as manager of peace-making for the whole Arab world.