The Arlington County Board approved a cut in the real estate tax rate yesterday that promises to save the average home owner $13 on his tax bill this year despite sharply higher assessments that have boosted the value of the typical home in the county by 15.1 percent.

The new rate -- down to 96 cents per $100 of assessed value and 16 cents less than figures used in the fiscal 1981 budget -- raised fears among the board's liberal faction that some local programs affected the Reagan administration's cutbacks in federal spending may have to be abolished and that school officials also will be forced to tighten their belts.

In other actions yesterday, the board approved a controversial condominium high-rise in the Rosslyn area, bringing some relief to county planners who long have ruled the lack of residential construction in the rapidly developing commercial center.

The board's fiscal action continues a conservative trend epoused most forcefully by former board chairman Walter Frankland, a leader of the board's Republican majority. It also represents a compromise between the board's three GOP members and the board's two Democrats.

Frankland and Board member Dorothy Grotos asked for much sharper cuts in the tax rate than the board was willing to vote. "The only way you can get this thing [spending] under control is to provide less money for the [county] manager and his staff's disposal," Frankland said.

"I've received no complaints that services are dropping off" as a result of previous cuts, he added. "In fact, I detect an improvement in some of the services that we get."

Newly elected board member John Milliken, a Democrat, argued, however, that the county should be prepared for severe cuts in federal assistance and predicted that the lowered property tax rate, on which the county budget must be prepared, may threaten the existence of some services that depend on federal support.

The new rate should leave the county with about $1.5 million in unallocated revenue, an amount that could be used to recoup losses of federal aid being proposed by the Reagan administration. The school board's requested budget, however, is already $3 million higher than the level that the county board has indicated it would fund. Yesterday's vote was another indication that the county board and the school board are on another collision course over education spending this year. s

Milliken dissented yesterday over the way the board was setting the county's tax rate before it knew what programs the county could afford in the coming fiscal year. "This is a backwards way to go about setting a tax rate," he said. "You have to tell citizens what programs you plan to cut and what deletions you're going to make."

"It's a good thing to ask," Frankland responded. "I'm not going to answer it at this time. I'm willing to start all over on the budget."

The new lower tax rate passed despite objections from Milliken and his liberal colleague Ellen Bozman. Board Chairman Stephen H. Detwiler supported the reduction, but said he could not back a tax cut as deep as the ones his Republican colleagues wanted.

The board also grappled with the long standing problem of providing places for people to live in Rosslyn, which has developed almost exclusively into a commercial and office center that some have dubbed a "concrete desert."

The board considered two proposals for condominium high-rises. Both called for 18-story luxury high-rises -- six stories over the limit generally allowed in the area. One was approved, the other rejected.

The board approved plans from International Developers Inc., one of Northern Virginia's largest condominium developers, to build 503 units on Fairfax Drive overlooking the Potomac despite complaints from some residents of nearby neighborhoods who said that the development would cause parking problems and would block their vistas of Washington.

Board member Grotos chided county planners for what she described as a lack of consistency in administering plans for Rosslyn's growth. "It seems to be the exception rather than the rule that developers come in and don't pay any attention to the plan," she said, referring to the county's policy of allowing builders to exceed height limits in return for concessions such as street and park improvements.

"It doesn't seem that there's any planning to it any more."

The board rejected a proposal to put up adjacent to the Colonial Terrace neighborhood a condominium almost identical to that planned for Fairfax Drive. Dozens of neighborhood residents came to the board to show opposition to the building. Board members also indicated they were disturbed by the "mass and bulk" of the proposed structure. They denied necessary zoning amendments.

One of the things offered by the developer for height concessions was a donation of 48 apartments as low income housing.While rejecting the project, board members asked the developer to consider scaling down the building and submitting another proposal that would preserve the intent of the project.