I offer what is claimed to be the most official record of this incident, from an internet forum:
I will now gave a list of his medal-awards from public
obituary in "Glas Crnogoraca No.16 April 16. 1911."
awards:Knightly order of St.Anna,I and II class,for wining in duel
against Japanese officer.For gallantry and bravery.
Wound medals for being wounded:First medal for being wounded in duel
against Japanese officer at Jun-ju-an.
Jun-ju-an (Sōngshùshān, 松樹山) was the name of a battery at Port Arthur.
The linked thread additionally contains heavy documentation that there were indeed one-on-one duels between Russians and Japanese. It has a newspaper account of a Japanese defeating a Russian. The forum posters are inconclusive on whether or not Saičić really dueled a Japanese, due to lack of documentation other than his obituary.
However, somewhere down the line the alleged duel has morphed into a tall tale:
Armies of two great empires with the greatest interest watched
heroes, who will share the duel in front of them . The surreal
setting, the heroes are approaching each other. with breakneck speed
they are clashing their swords dozens of times It was a clash of two
traditions and two methods of warfare. One swing of the samurai katana
grazed Saičić's forehead.
Japanese warrior,who saw the blood on
Montenegrin was determined to end the duel. In that moment Lekso
Saičić with his lightning speed waved the sword and behead a Japanese
hero. As soon as he fell to the ground, samurai horse ran off back
Japanese army. By this victory Saičić applied a heavy moral defeat the
Japanese army, which has caused disappointment, but joy and happiness
to the Russian army . moving back from the duel ground, Lekso as a
true knight, paid tribute to deceased Japanese soldier.
approached the Russian army, lined up in his honor, he gave a command:
QUIET! Military music was playing military march. The commander of the
Russian army generals, met the winner of the duel. Lekso greeted the
commander and reported to him that he had performed his task. The
commander greeted him back and congratulated him the victory, as well
as the other generals. Then they treated him with highest military
honors. Admiral of the Russian Fleet Rodzenstvenko and the Japanese
Fleet Admiral Togo also congratulated him.
First of all, there is no record of this event in Japanese, so I have to write this post from the frustrating perspective of proving a negative. But basically, this retelling is completely silly.
The samurai part is historically impossible, because samurai were abolished with the declaration of the Meiji Restoration in 1868. The Russo-Japanese war was not a place where people would put on old costumes and go dueling.
There was no incident in the Russo-Japanese War where the armies simply stood and watched two of their warriors fight. At Port Arthur, the batteries were constantly firing down on the Japanese. There was no time to pause and wait.
The katana part is not possible either. Officers in the pre-fascist era Imperial Japanese Army were issued cheap ceremonial swords of European style, not samurai swords. Some also had sabres, but these would have been more clearly identifiable.
The Togo part is ridiculous. Tōgō Heihachirō became well known throughout the West because he led the small Japanese Navy to a shocking victory against Russia's Czarist Fleet. But he had nothing to do with Port Arthur, and it would have been highly improper for him to congratulate someone who defeated a Japanese officer in another wing of the armed forces. Such an event would have created a scandal in Japan.
Basically, I cannot answer whether or not there was a sword duel or a scuffle, but the "samurai duel" version is definitely false.