Badly punished but scarcely destroyed, Christian East Beirut began digging out yesterday from the rubble of the worst shelling it has taken since violence erupted in Lebanon 3 1/2 years ago.

Both the Christian militia and Syrian artillery - which caused almost all the extensive damage during the week-long bout of fighting - were honoring a precarious cease-fire more than 24 hours after it took effect Saturday night.

Syrians declared the cease-fire unilaterally in response to a U.N. Security Council appeal issued after strenuous U.S. and other Western diplomatic efforts. Washington feared the furious Beirut fighting could result in open Israeli military intervention against Syria and jeopardize the Camp David agreements with Egypt.

But at least four Christians were killed by Syrian machine gunners as they drove across one of the key Syrian-controlled bridges trying to leave East Beirut Sunday morning. The Syrians, who first allowed traffic in and out of East Beirut, closed the road for a time but then again began letting cars leave - after careful searches. But they barred cars from entering the inner Christian stronghold.

Destruction in the suburbs of Dora and Ddaideh and in Bourj Hammoud an adjacent-Armenian suburb across the Beirut River from East Beirut, was reported much more extensive than in the city itself.

It was there that Syrian gunners concentrated fire to prevent Christian militiamen from threatening their troops overlooking bridges controlling the approaches from the Christian hinterland to East Beirut itself.

Inside East Beirut, destruction varied enormously. The worst punishment appeared reserved for the rubble-strewn bridges' approaches - although the bridges themselves were largely untouched. Also hard-hit were headquarters of the various militias and areas adjacent to a Syrian strong-point atop an unfinished skyscraper.

The Phalangist headquarters near the port had its facade destroyed by two 240 mm mortar rounds, but none of the 20 Phalangists inside at the time was even scratched, party officials said.

The area around National Liberal Party leader Camille Chamoun's headquarters also was seriously smashed with houses and upper stories collapsed and facades full of holes.

But whole neighborhoods - and even entire streets close by the worst destruction - were spared or escaped with only broken glass or an occasional artillery, rocket or mortar round through an upper story.

In some places telephone and electricity had functioned intermittenly throughout the nine-day bombardment - sometimes better than in virtually untouched West Beirut. In many areas, however, city water was cut off and power and telephone lines dangled overhead.

Some inventive East Beirutis attached branches to their cars' bumpers in a homemade remedy to sweep away glass littering many streets.

Official spokesmen for the Phalange and National Liberal parties talked confidently - and said their men were ready to start fighting again if their demands for total Syrian withdrawal from Lebanon are not met immediately.

Chamoun, a former president, denounced incumbent President Elias Sarkis as a "traitor" and "quisling" for having negotiated the cease-fire with Syrian president Hafez Assad in Damascus.

Other militia spokesmen conceded that Sarkis had succeeded in strenthening his own weak position at their expense in the Damascus talks.

But haggard, near hysterical residents of East Beirut were far from sharing such official optimism. Correspondents watched dozens of heavily laden cars brave Syrian machine gun fire in desperate - and sometimes fatal - attempts to flee East Beirut across the bridges for the Christian hinterland.

Throughout the fighting, discipline and experience helped limit the casualties. East Beirut has been targeted for so long that its remaining residents - perhaps 20,000 to 30,000 of the normal 300,00 population - went automatically to well stocked shelters.

With residents only beginning to dig out, even Dr. Eli Karameh, the Phalange medical director and a party Politburo member, stressed the casualty figure of 80 dead and 3,000 wounded was approximate.

He said the militias lost between "70 and 100" men. Of the total civilian casualties, half were treated in neighborhood dispensaries for minor wounds and the others had been hospitalized from anything from "one to seven days," he said.

Dr. Karameh said underground springs had been found underneath apartment building. His first aid men had risked their lives tirelessly to keep the shelters supplied.

"That is why we did not have a great catastrophe," he said, "for otherwise many people would have died for lack of food - and especially water."

East Beirut hospitals have functioned with underground operating theaters and beds were placed in corridors.

Wadieh Wakim, a 27-year-old accountant, showed visitors where he and 15 others had taken shelter in a ground-floor apartment on a particularly punished street near one of the Beirut bridges.

"I know it sounds trite, but it was a real hell," he recounted, "with the bombardment especially intensive the first three days and during the next-to-last night, when there was no letup around the clock."

Without city water or electricity, he and the others survived on their stocks of food and bottled mineral water. They listened to their transistor radios - "the BBC, Monte Carlo, the Voice of America, the Israel radio - anything - but news, only news."

In the street a dead dog and a dead rat testified to the violence of the pounding the neighborhood had taken.

Yet at the crossing point dividing Christian and Moslem Beirut, the shooting of the last nine days was only a memory - once again. A virtual shooting gallery in bad times, the area was peaceful, with the more than a dozen taxis waiting to take Christians to West Beirut.