Fierce winter weather and the full impact of the national recession delivered a one-two punch to Virginia in January, sending the state's unemployment rate to a post-World War II high of 7.8 percent.

But for those who are out of work, there was an encouraging word: Gov. Charles Robb announced today that the state will reopen 13 branch offices and restore placement services at 12 other locations.

The Virginia Employment Commission's Jan. 10-16 survey showed 202,400 Virginians without jobs-- 35,000 more than in mid-December, when the rate was 6.4 percent.

The jobless rate was the highest since it hit 7 1/2 percent in March 1975, and 1.7 percent higher than January 1980. Things were worse in some of the state's metropolitan areas, where layoffs were concentrated.

The Norfolk area's jobless rate soared 1.7 percentage points to8.4 percent in January. Roanoke's rate was 7.1 percent in December, and rose to 7.7 percent in January.

Northern Virginia continued to have the lowest unemployment rate of the state's metropolitan areas: 4 1/2 percent. But even in that populous area, home for many federal employes who work in D.C., the rate was up from 3.9 percent in December.

VEC spokesman William Mezger noted that unemployment normally rises in January because of winter weather, which prevents outside work, but said this particular January was especially vicious. At the same time, the national recession "was beginning to take hold" in the state, he said, adding:

"Nearly all factory categories experienced losses in January, but the auto and auto component, furniture, textile, apparel and building material segments were hardest hit."

About 72,000 Virginians were drawing unemployment compensation in January, the VEC said, or 10,000 more than in December and in January 1980.

The jobless figures were below the national rate of 9.4 percent.

Restoration of the VEC branch services that had been cut depends on whether the host cities will lease office space and pay utility bills for those branches, officials said.

VEC branches were closed in Arlington, Chesapeake, Front Royal, Hampton, Manassas, Pulaski, South Richmond, Salem, South Boston, Suffolk, Virginia Beach, Williamsburg and Wytheville. Discontinued placement services will resume at Alexandria, Buena Vista, Covington, Culpeper, Emporia, Farmville, Galax, Marion, Martinsville, Portsmouth, Richlands and Warsaw.

Sen. John Warner (R-Va.) cosponsored a supplemental appropriations bill that last week gave the states$2 billion to replenish jobless funds and restore employment services hurt by congressional budget cuts. Virginia's share was $2.8 million.