The Reagan administration lifted its controversial 76.4-mile limit on Metro subway construction yesterday and agreed to turn over $24 million to the transit authority to start building rail extensions in Prince George's County and Alexandria.

The federal grants, which had been held up for a year by the administration, were released as a result of protracted negotiations and congressional pressure. "This is a show of good faith on the part of the administration," said Ralph L. Stanley, the administration's mass transit chief.

The grants, announced at a news conference on the Capitol steps, will allow Metro to begin work on a Green Line station at West Hyattsville in northern Prince George's and a Yellow Line spur to a Van Dorn Street stop in Alexandria's West End. The extensions are scheduled to open in the early 1990s.

The federal government's approval of the two projects was hailed by Washington-area officials as a key step that is expected to lead to carrying out Metro's long-debated plan for expanding the subway system to 89.5 miles. The 89.5 miles would be financed with funds already authorized by Congress.

Three months ago, federal officials approved the main outlines of the 89.5-mile plan "in principle" and began negotiations with Metro officials over a contract to put it into effect. Recently, however, the plan has appeared in jeopardy because of Senate and administration moves to cut spending for Metro.

With yesterday's agreement, the administration now has endorsed more than 82 miles. The current system is 60.5 miles. An Orange Line extension to Vienna in Fairfax County is to open next year. Work also has begun on the Green Line between Anacostia and Northwest Washington and on a Red Line link to Wheaton.

Metro's 89.5-mile plan would provide for one additional Green Line segment in Prince George's, connecting West Hyattsville with Greenbelt. Metro's long-term plans call for 103 miles, but no funds have been earmarked for the last 13.5 miles. The administration opposes any further congressional aid.

For Prince George's, the West Hyattsville project marks the first subway construction in the county since the Blue Line's Addison Road terminus opened nearly five years ago. County officials have repeatedly called for stepped-up work on the Green Line.

The West Hyattsville station, near Ager and Queens Chapel roads, is scheduled to open in late 1992 as part of a Green Line branch linking the Fort Totten station in Northeast Washington with the Greenbelt terminus. Construction is expected to start in the fall.

In Alexandria, the new grants provide funds to build three Metro bridges over Eisenhower Avenue and Cameron Run. The bridge work will be an initial step in construction of the four-mile spur linking Alexandria's King Street and Van Dorn Street stations. The extension is to open in late 1990.

The administration had refused to release the grants last June, criticizing construction plans as "fiscally imprudent." Metro's 89.5-mile plan, drawn up in September, was aimed partly at convincing the administration that the authority's proposals were financially sound.

At yesterday's news conference, Stanley was flanked by Maryland and Virginia senators and congressmen along with Metro officials. Sen. Charles McC. Mathias Jr. (R-Md.) said the grants "should set us on our way toward the full" 103 miles. Sen. John Warner (R-Va.) termed the action "a major breakthrough."

Rep. Steny H. Hoyer (D-Md.) said that the administration's agreement "means they have backed off attempts to limit" Metro's expansion.