For his first act as lieutenant governor of Virginia, Charles S. Robb, the new hope of the state's Democratic Party, will lead his supporters in a festive effort to help pay off $400,000 in debts piled up by the Virginia Republican Party and its statewide candidates.

The son-in-law of the late President Johnson has not suddenly decided to play both sides of the street in this ticket-splitting state. He is merely the captive of an inaugural celebration that embraces the winners of all three statewide offices but benefits only the party of new governor. The new governor will be Republican John N. Dalton.

The Dalton Inaugural Committee has included Robb in its palns for the Jan. 14 celebration and for invitations to the round of balls that would be held in the capital city that night.

To attend, those invited must pay from $70 for an individual who declines the post-ball breakfast to $150 for a couple desiring breakfast. Republican National Committeewoman Judy Peachee, the ball director, estimates her party will gross about $300,000 from the event and net about $200,000.

The way the Republicans figure it, the Robb followers could contribute as much as $16,000. It's not a lot, but the GOP is pursuing every dollar it can get.

Not only will Robb be helping Dalton and his party pay its debts, he also will be giving a hand to the candidate he defeated, state Sen. A. Joe Canada of Virginia Beach. During the prolonged heat of the campaign, Canada dismissed Robb as someone who would not have a chance had not he married Lynda Bird Johnson, daughter of the late President.

Moreover a slice of the inaugural profits will go to the new attorney general, Republican J. Marshall Coleman. The compulsive matchmakers in the press and party organizations already have paired Coleman and Robb as the likely opponents for governor in 1981.

Coleman needs to pay some campaign debts of his own before gearing up to run against Robb and presumably will accept help from any quarter.

Robb aides are dismayed by this set of facts but are determined to proceed gamely with the inaugural arrangements. Sandy Duckworth, Democratic national committeewoman from fairfax County and the director of Robb's inaugural activities, said in an interview:

"I nearly died when I learned about this of course I don't like it one bit. But I don't think there is anything we can do about it. Someone told me this is the way it has always been. I guess we would be doing the same thing if we were in Dalton's situation.

"They have been very cooperative with us about the arrangements and invitations and of course we are being very cooperative with them."

Robb has the biggest campaign debt of all the statewide candidates in the Nov. 8 election. He spent a record $725,000 to win the part-time lieutenant governorship and owes $218,000, compared to $200,000 for Dalton. Dalton spent a record $1.9 million for his campaign.

Duckworth was in Richmond today making plans for a separate Robb fund-raiser to be held during the inaugural weekend.

There is precedent but hardly a tradition for the Robb dilemma. Republican Gov. Linwood Holton staged the only previous fund-raising inaugural ball in 1970 with the participation of a new Democratic lieutenant governor and Democratic attorney general. Holton's party got all of the $50,000 profits.

In 1974, Republican Gov. Mills E. Godwin, a former Democrat, returned to a nonpartisan celebration. Ball tickets went for a top price of $35 and the $13,000 profit was donated to charities.

The Democratic pain is made more acute this year not only by the higher ticket prices and the prospect of higher GOP profits, but also by the recent Republican ascendancy in statewide politics.

Holton was the first Republican governor since Reconstruction. Today Virginia is the only state in the nation that has not elected a Democratic governor or senator in the last 10 years, and Democrats blame the solid financial of GOP candidates for much of their decline.

As usual, the official inaugural ceremony on the Capitol grounds will be free. By law, it is conducted by the General Assembly.

In addition to the nightime balls following the inauguration, the Dalton Committee is planning seven other celebrations throughout the state during the inaugural period. SOme parties with the partial goal of raising money and some are not.

In Danville, where voters recently bent down yet another attempts to legalize liquor by the drink, the Dalton victory will be celebrated with a prayer breakfast.