The Metro board voted yesterday to apply for the first federal grants for its four-year, $1.8-billion building program, amid fears that President Reagan's call for a further 12-percent reduction in federal spending could undo the plan before it starts.

The plan, which depends almost entirely on federal funds, provides for all segments of the complete 101-mile subway system to be operating or under construction by 1985. Metro devised the plan in an attempt to get construction under way quickly in case federal funds run short in coming years.

But Reagan's proposals last week for new cuts immediately indicated that the funding challenge will come sooner than expected. Yesterday, the anticipated cuts were condemned strongly after the board voted to apply for $315 million to finance construction, design and car purchases in the plan's first year.

Board Chairman Joseph Alexander called for energetic lobbying against the cuts, which he said could bring "the death of transit as you and I know it in this country." The transit authority's general manager, Richard Page, called the cuts "an unannounced plan to dismantle the federal mass transit programs."

Last month, the House approved a level of $315 million in capital grants for the transit system. Until Reagan's statement, Metro officials had been confident that that sum was assured. The Senate has yet to act.

If Congress does agree to a 12-percent rollback, it is not clear how grants to Metro would be affected. Page said that one proposal being discussed in the Urban Mass Transit Administration, which administers the grants, is to reduce capital outlays by as much as 25 percent so as to soften the blow on cuts in federal operating subsidies.

Many of the nation's transit systems depend on these subsidies to fend off bankruptcy. However, federal subsidies cover only about 8 percent of Metro's operating costs and it opposes concentrating the cuts on construction grants.

If building funds are reduced, the four-year plan provides for a pro-rata reduction in construction programs in all eight jurisdictions that are members of Metro, so that no one jurisdiction would lag behind the others. Officials in Prince George's County and the District, where most of the unstarted track is planned, worry that each year that construction is delayed increases the likelihood that the track will never be built.

In other business, the board:

* Authorized Metro staff members to explore issuing short-term bonds. Up to $25 million could be raised this way, finance chief William Boleyn told the board. The money is needed to cover cash shortages caused by late payments of subsidies, Boleyn said.

* Presented Peter Kiewit Sons' Co., a construction firm excavating the Red Line station at Forest Glen, with an award commending it for 500,000 "man hours" of work performed without a single accident serious enough to take a worker off the job.