The Alexandria school board, facing major reductions in federal funds if President Reagan's proposed cuts are approved, last night adopted a $44.5 million school budget for the next school year.

No additions to the already proposed 10 percent spending increase over the current year's $40 million budget were included to offset the possible federal cuts, but school officials mapped a campaign to head them off.

"There is no doubt that the [proposed] cuts readlly hurt urban schools," Superintendent Robert W. Peebles said in an interview before the vote. "However, we have no plans to ask for additional increases than those already included in the proposed budget. Instead, we plan to work with Fairfax and Arlington County and our elected representatives in fighting the cuts."

The budget may face stiff opposition from the City Council, which had asked all city bodies to limit spending increases to eight percent.

The proposed federal cutbacks could mean heavy losses for the 11,000-student school system's lunch and special education programs. Estimated losses could run as high as 35 and 20 percent, respectively, of previously anticipated federal funding for these programs.

Proposed cutbacks in so-called impact aid to school districts that have many children of government workers also could cost the school system half a million dollars. o

"It's ironic. The proposed cuts are intended to bring back more control to the local level, but really they only increase the burden on the local community," Peebles said.

Although school officials were unwilling to specify exactly how much they expect Reagan's cutbacks to cost, school officials earlier had said the $1.75 million lunch and breakfast programs could be among the hardest hit. Nearly 50 percent of all Alexandria students receive either free or reduced-cost meals. Under the present formula, the federal government provides nearly half -- $863,000 -- of the funding for the program. That could be reduced by more than a third if Congress accepts Reagan's proposal.

School officials said prior to the hearing that the system's special education programs could ber seriously jeopardized by proposed changes in federal funding and restrictions. Heavily enrolled, remedial reading programs and classes for the handicapped are among the 57 federal school-aid programs recommended for a 20 percent slash and consolidation under two large "block grant" programs. In Alexandria, these two programs alone serve more than 2,200 students.