Sen. Warren B. Rudman (R-N.H.) went to the White House on Wednesday afternoon skeptical not only of the MX missile but of President Reagan's general commitment to arms control. He sat beside the president for 30 minutes of earnest conversation and came out of the Oval Office a believer.

The transformation of Rudman from opponent to skeptic to supporter of the MX provides an insight into how Reagan and his lieutenants won two big victories on the controversial missile this week after suffering one setback after another to their strategic blueprint last year.

"I told White House lobbyist Ken Duberstein that I really thought it was important for the president to understand why I thought the public perception of him on arms control was not right, that it would be good for me to talk to him directly," Rudman said in an interview last night after voting with the president as the Appropriations Committee, by 17 to 11, approved $625 million for the MX.

Shortly after Rudman explained in a Washington Post article of April 30 why, even as a pro-defense Republican, he was having trouble going along with Reagan on the MX, Duberstein called to tell him the Reagan meeting was in the works.

The administration and its allies on the MX, realizing that the undecided Rudman's vote could be decisive on the Appropriations Committee, paid a great deal of attention to him in the days preceding the meeting with Reagan.

Brent Scowcroft, the retired Air Force lieutenant general who headed the presidential commission that recommended putting 100 MX missiles in Minuteman missile silos while developing a smaller mobile missile, stopped by Rudman's office on Monday for a 90-minute chat. Ronald Lehman, an articulate and genial assistant secretary of defense who specializes in nuclear weapons, called on Rudman on Tuesday. He spent 45 minutes answering the senator's questions and making the case for the MX package.

Then there were lengthy telephone conversations with former secretary of state Henry A. Kissinger that Rudman instigated.

Despite all this, Rudman said he was still leaning against voting for the MX. He just did not see how putting 100 MX missiles in vulnerable Minuteman silos was worth the estimated cost of $18 billion when the new, small missiles would be coming along a few years later.

But Reagan in person, and the letter the president wrote to three other senators committing himself to a "build-down" of nuclear weapons by scrapping old ones as new ones were deployed, converted Rudman.

"I started out by telling him I was from Nashua, N.H.," Rudman recounted. "'You remember, Nashua, Mr. President,' I said, 'you bought a microphone there.' He laughed at that and told me to sit down in one of the two wing chairs standing side by side in his office. He sat in the other one. Duberstein was the only other one in the room.

"I told the president that I was very concerned about the administration's position on disarmament and said that many people in New Hampshire didn't think the administration was as serious as it should be. I also told him that some of the members of his administration had made statements about nuclear matters that had scared people and driven them into the freeze movement.

"I told him, 'These are good people, Mr. President. My friends, your friends. I go home every weekend. I know how they feel. I have an obligation to tell you that they they don't think this administration is serious enough about arms control.'

"He seemed a little stunned after I said that. With real vehemence, looking me straight in the eye, the president said: 'Anybody who says I'm not interested in disarmament is just plain wrong. I'll do anything that will lessen tensions and maintain our national security.' "

Rudman said that at another point in the session Reagan said one reason he needed to deploy the MX was that everyone who had wrestled with the Soviets on arms control had told him, " 'The only way you can get anywhere with the Soviets is to have something to negotiate with; if you're going to get something, you have to have something to give.' "