Sunday subway service, the missing link between Metro and legitimacy as a full-fledged big-city system, appears to be gaining adherents as the Metro board nears final action on a new fare-and-service schedule.

At a board committee meeting yesterday, the Metro staff presented figures showing that week-night and Saturday service-inaugurated last September and October-has been more successful in attracting riders and their money than was projected.

Based on that experience and the relatively small additional subsidy needed to operate on Sunday, the staff strongly reiterated its recommendation for Sunday service starting in September.

Board members polled informally and on a not-for-attribution basis yesterday said they believed there were four solid votes for Sunday service, including both District of Columbia votes and one each from Virginia and Maryland. That is all it would take to approve the service, but there is a problem.

The Metro Board is traditionally reluctant to adopt measures the cost money without a unanimous vote of six because each Washington area jurisdiction that participates in Metro contributes voluntarily to the subsidy be paid, though it always has been.

It will cost an estimated $800,000 more in subsidy in fiscal 1980 to provide eight hours of Sunday subway service. The total operating deficit for 1980 is projected at $120 million, although that is expected to be reduced somewhat by a fare increase scheduled for July 1.

Prince George's County and Arlington County, both of whose governing bodies have voting members on the Metro board, earlier have indicated opposition to subsidizing Sunday service.

"I don't think there are three votes on the (five member) Arlington County Board," member Walter Frankland told the Metro board yesterday.

Prince George's County Executive Lawrence J. Hogan has recommends against Sunday service. Council member Francis B. Francois, also a member of the Metro board, said "I don't know" when asked what the County Council would do.

Both Arlington and Prince George's politicians are under pressure to reduce expenditures.

During a recent round of public hearings, almost all the testimony on the question favored Sunday service.

The board asked yesterday for cost figures on 12 or 13 hours of service as well as the eight hours the staff has proposed.

Saturday service is now averaging about 72,000 riders per Saturday-slightly more than the Staff projected. Weeknight service-probably form 10 a.m. to 6 p.m.-would draw about 30,000 riders, the staff-projects.

In a related matter, the Metro Board was told yesterday that a one-week record of 1,399,795 riders boarded Metro's trains the week April 16, including 83,842 riders on Saturday.