The Fairfax City Council voted unanimously last night to approve a $38 million budget for fiscal 1987 that calls for cutting 1 cent from the real estate tax rate instead of raising it by 3 cents as City Manager Edward A. Wyatt proposed last month.

The council's action drops the city's real estate tax rate to $1.12 per $100 of assessed value. However, most homeowners will receive higher tax bills because of the increase in the assessed value of their properties.

For instance, the owner of a house assessed at $95,800 -- the median value of houses in the city -- will pay $1,073 in taxes for the fiscal year that begins July 1. The same owner's last tax bill was $39 less, based on an assessment of $91,500.

The budget includes a 4.5 percent spending increase from this year's $36.4 million spending program.

The council approved a 4 percent cost-of-living raise for all city employes and recommended 11 new full-time positions, which include city bus drivers, a patrol officer, a police dispatcher and an mechanic for city vehicles.

Council members allocated $12.6 million to the city's school system to cover an estimated $10.8 million in tuition fees to Fairfax County, which operates the city's six public schools.

Next year's school budget now includes a 43 percent increase in funds for renovations and improvements to the school buildings, which are owned by Fairfax City. These improvements include $240,000 for a multipurpose room addition at Jermantown Elementary School, $94,000 for a music room addition at Lanier Intermediate, and $125,000 for a home economics room and design fees for a new gymnasium at that school.

The council also appropriated $816,296 to its transportation fund, which will cover the operating costs of the city's new local bus service to the nearby Vienna Metrorail station. The service is scheduled to start when the Vienna station opens June 7.