There could be a group of rogue shinobi and demons who have joined forces. They see the Juubi as a source of ultimate power and plan to use it to reshape both the Naruto and Inuyasha worlds according to their own desires. This group would be a complex antagonist as they have different abilities from both universes and would require Naruto and Inuyasha to use all of their skills and the combined powers of their friends to defeat.
It could be a powerful ninja from Naruto's world who has been corrupted by the Juubi's power and somehow ended up in the Inuyasha world. He wants to use the power of the demons in the Inuyasha world to make himself even stronger and take over both worlds.
Madara Uchiha could be a great antagonist in this fanfic. Even though he was defeated in the original Naruto story, if he somehow gets resurrected and gets a chance to control the Juubi, he would be a formidable foe. He already had grand plans for the world in the original story, and with the Juubi's power, he could be even more dangerous. He might want to reshape the world according to his vision, and Naruto would have to fight him once again to protect everyone.
The plot could involve the Juubi's chakra influencing some of the demons in the Inuyasha world. Naruto arrives and has to figure out how to stop this chaos. He might teach Inuyasha some of his ninja skills, and Inuyasha in turn could show Naruto how to deal with the demons. They could also find that the Juubi has some ancient connection to a powerful artifact in the Inuyasha world, and they have to stop an evil force from using this connection for their own gain.
It could be a powerful ninja - demon hybrid. This antagonist could be jealous of the bond between Naruto and Inuyasha. He wants to use their combined power for his own evil purposes. So, he tries to turn them against each other first by spreading false rumors and then by attacking their loved ones.
Perhaps a rogue Sekirei who has been corrupted by the power of the Juubi's energy that has seeped into the Sekirei world. This Sekirei could start causing chaos, believing that they are now more powerful than the others and should rule over them. Naruto would then have to stop this Sekirei from causing more destruction and bring balance back to the Sekirei world.
A common theme is the clash of different power systems. Naruto has his chakra and various jutsu, while Inuyasha has his demon powers. When the Juubi is added into the mix, it creates a complex power dynamic.
It could be an evil ninja from Naruto's world who somehow gains Kryptonian - like powers. He becomes power - hungry and decides to take over both the Naruto world and Supergirl's world. His knowledge of ninja techniques combined with super strength and other abilities would make him a formidable foe.
Madara Uchiha would also make a great antagonist. Imagine if he was resurrected and saw Naruto with Samehada as a threat to his plans of creating a new world order. He would use his Sharingan and Rinnegan powers to try and take Samehada from Naruto, believing that with it, he could easily defeat all who oppose him and reshape the ninja world according to his will.
Maybe another kitsune who has gone bad. In the fanfiction world, there could be a rogue kitsune who sees Naruto as a threat. This antagonist kitsune might be jealous of Naruto's power or his connection to the human world. They could try to undermine Naruto's attempts to protect the village or his friends.
It could be a powerful dark spirit from Kagome's world that somehow manages to cross over to the Naruto universe and starts causing chaos. This spirit might be immune to normal ninja attacks and require Kagome's spiritual powers to be defeated.
Perhaps an ancient and evil ninja from a long - lost clan that was sealed away behind the gate. He resents the modern ninja world and wants to destroy it. He has unique and powerful jutsus that Naruto and his friends have never seen before.