The Austin City Council today began a unique venture in salesmanship: disposing of a share of a nuclear power plant.

On Tuesday, Austin voters decided to end the city's participation in the South Texas Nuclear Project, which has been plagued by delays and cost overruns.

Nuclear industry experts can recall no other city in the United States that has ever voted to back out of an existing nuclear project, and the decision has left Austin officials with some serious practical problems.

The first is finding a buyer for the city's 16 percent share of the 2,500-megawatt project, scheduled to come on line in late 1986. The second is finding an alternative source of electrical power for one of the fastest growing cities in the Southwest.

It could take the city more than a year to get out of the plant, assuming that it can find a buyer, and it is possible that the residents of Austin may take a financial beating.

The other three partners in the project have first right of refusal. Houston Light & Power, the managing partner, is strained for credit. The city of San Antonio has reservations about its 30.8 percent share of the project and isn't looking to enlarge it.

Central Power & Light Co. of Corpus Christi is saying nothing. If none of those cities is interested, Austin must look to other utility users, presumably in Texas.

Austin utility customers have invested about $300 million, and have authorized bonds that will bring the amount to about $375 million by mid-1982. The entire project was originally supposed to cost $1 billion, but current estimates put the final tab at between $4.4 billion and $4.8 billion.

Unless Austin finds a buyer soon, it may lack the money to invest in new energy resources. The city grew 36 percent between 1970 and 1980, and its energy needs grow about 6 percent a year.

Likely alternatives appear to be Texas lignite, a low-grade coal, or western coal. Both are expensive.

Austin is considering whether to participate in a three-way bid for lignite reserves at Camp Swift, but the deal is clouded by U.S. threats of an antitrust suit.

Anti-nuclear activists on the City Council prefer a massive energy conservation program and experimentation with solar power, rather than coal or lignite.

Houston Light & Power officials said Austin's vote will not affect the future of the plant, and seemed to delight in the prospect of seeing the anti-nuclear faction in Austin being forced to persuade someone else to buy into the plant.

"I don't mean to sound snippy," said Don Beeth, a spokesman for Houston Light & Power, "but it's their problem."