The Alexandria City Council yesterday approved a $131,000 program that will provide a court psychologist, family counselor and overnight shelter for juveniles arrested in the city.

Juvenile and domestic relations Judge Joseph L. Peters Jr. appeared before the council to endorse the Juvenile Justice Program, the first of its kind in the city. "It's important to have these services," said Peters, who adjudicates all of the city's juvenile cases.

Peters has said in the past that he often sees the same 13-year-old who appeared before him on a petty charge return as a 15- or 16-year-old charged with a more serious crime.

The new program will focus particularly on the needs of first-time offenders under age 18.

The judge said he made his unusual appearance before the council yesterday "to persuade them to follow through" on the program's funding.

"This is tremendous. There is a real need for services that focus on preventing, and not simply punishing," Virginia Hines of the city's Youth Services Commission said yesterday.

Hines said she will recommend by next month possible sites for the temporary shelter for youths detained on status offenses, or juvenile offenses such as violating curfew and running away from home.

"Until now, there was no place for the youth who ran away from home," Hines said. "You don't want to throw him in jail, and maybe he can't go home."

The council voted unanimously to allocate $33,000 for the shelter. Hines said she did not know how many youths could be housed on that budget.

There are 180 juveniles on probation in Alexandria, according to a city report. All of those youths will be entitled to the services of a court pyschologist, a family counselor and a special projects coordinator provided by the council's unanimous vote yesterday.

The special projects coordinator will coordinate community and business involvement in an effort to counsel, teach and employ young offenders.

"If the program works, it'll save a lot on crime," said council member Donald C. Casey. "Many of the break-ins and auto thefts are committed by juveniles."

In 1983, the last year for which statistics are available, police said, there were 2,400 complaints filed against juveniles.

In a separate vote for program funding yesterday, the council moved one step closer to approving a $67,000 expenditure for an adult day care center.

The center, which would provide an organized activities program for elderly persons who are physically disabled or mentally impaired, must be approved at the April budget hearings.

"There is no question that there is a need for this," said council member Robert L. Calhoun. "We'll have to come up with the money somehow."

The 65-and-older group in Alexandria is the fastest growing sector in the city, said Mark Horowitz, deputy director of Human Services. About 10 percent of the city's population, or 9,465 Alexandrians, are over the age of 65, according to the 1980 census.

"There is a definite need to bridge the gap between home care and expensive institutional care," said Kathryn A. Larmer, the chairman elect of the National Institute on Adult Day Care.

"Until you know what it's like to take care of someone around the clock, you don't know how desperately we need a place like this," Irene Y. Bowen, 68, told the City Council.

Bowen, a legal secretary who retired last year because her husband Theodore could not stay at home alone, said a day care center not only would benefit aged persons but also would relieve their families of the burden of constant home care.