The Alexandria City Council unanimously approved a $136 million budget early today, increasing city spending by about 9 percent while leaving the real estate tax rate at the current $1.37 per $100 of assessed valuation.

Although the tax rate will remain the same, rising assessments will mean that the amount of taxes paid will go up. The owner of an "average" home, worth $100,000, will pay about $100 more. Over all, real estate assessments have risen 16.5 percent in the past year, with assessments for single-family homes and condominiums going up an average of 7 percent and apartment developments and undeveloped commercial and industrial land rising 40 percent.

The council eliminated from the budget a controversial trash collection fee of $24 a year recommended by City Manager Douglas Harman to raise $1.2 million. Instead, a fee will be imposed on commercial trash haulers.

Budget deliberations this year were complicated by the loss of about $2.8 million in federal funds occasioned by Reagan administration economies. When the discussion finally ended after midnight, the spending total for the fiscal year that begins July 1 stood at $42,000 over Harman's budget proposal.

Among the features of the new budget is a $1,250-year pay increase for council members, bringing them to $12,500 a year when the new council is sworn in July 1. Four council members voted for the pay raise, with members Carlyle C. Ring and Marlee Inman voting "no." Vice Mayor Robert Calhoun abstained.

"I don't think people ought to confuse martyrdom with effective public service," said Councilman Donald Casey, who voted with the majority. "The city gets the best I can give them. I don't think we need to sell ourselves short."

The budget also provides for a $5 increase in auto licensing fees for vehicles under 4,000 pounds. The fee will go from $20 to $25.

Among the seven new city jobs created by the budget are four parking enforcement officers and one additional cable television staffer. The budget also provides for a 5 percent increase in salary and benefits for city employes.

In a budget session that one city resident described as "unusually nit-picking" the council members wrangled over reductions in the school budget and their salary increases.

In the end, the school budget survived intact, with the council voting down a proposal to reduce the schools' contingency fund by $100,000 after hearing pleas from two school board members. One of those members, Shirley Tyler, said: "I think we've made some very difficult decisions already. I think we've shared the misery."

The budget, which will use more than $2 million in surplus from the current fiscal year, will delay indefinitely a $90,000 expenditure for a city recreation center, the Chinquapin Park project.