When Hawkeye examines the unconscious wounded soldier lying on his side and realizes it's Radar, Nurse Kellye (seen from the back) had already been standing over Radar. Nurse Kellye had been at the 4077 since Col. Blake's days and certainly would have recognized Radar, so she presumably would have alerted one of the surgeons that Radar was among the wounded.
Radar ostensibly goes to Seoul to lose his virginity, but this presumably already happened in The Army-Navy Game (1973) when he invited a nurse to the supply tent when they believed that they would die in the enemy bombardment.
There's no way Hawkeye would be allowed to perform surgery while being drunk or hungover however busy they were.
When the unconscious Radar is being examined in pre-op, Col. Potter says, "What the hell was he doing out there in a jeep anyway?" But as commanding officer of the 4077th, Col. Potter would have had to sign Radar's pass to go to Seoul, so Potter would have known exactly why Radar was "out there in a jeep."