The Prince George's County Board of Education unanimously approved last night a $337.4 million budget request for running the school system next year and sent it to County Executive Lawrence J. Hogan, who has already said the figure is too high.

The request is 9.1 percent higher than the current school budget. But because of declines in the percentage paid by the state and federal governments, the county's contribution would go up by 19.3 percent this year. Last year, the county spent 49 per cent of its budget on the school system.

Several board members spoke last night of programs they had unwillingly forsaken, and of inevitable cuts by the county executive. "I'm not fool enough to believe we are going to have all this budget funded," said Vice Chairman Angelo I. Castelli, before he voted.

Hogan must submit his own proposal for school and county spending to the County Council by April 1. Last year, Hogan cut $11 million from the schools' request, but the council restored $10 million. This year, the county executive predicts that the council will be unable to repeat that performance.

"It's a pretty grim situation for fiscal '83," County Chief Administrative Officer Kenneth V. Duncan said yesterday. "We don't have as much money this time around. Everybody's flexibility is therefore diminished, including the county council's."

The major increases in the school board's budget proposal are $23.5 million in salaries and benefits, $2.5 million to replace 12-year-old buses as required by law, and $1 million for special education personnel formerly paid from federal funds. Teacher salary raises will be tied to the cost of living, but are expected to average 7 percent.

The school system hopes to save $4.5 million through staff reductions and school closings made possible by falling enrollment. Eight schools are scheduled to close this June. Another $500,000 would be saved by making students pay for drivers' education.