Who and why is the woman in Tomoe's arms right before he goes on search of Dragon King's eye? In Kamisama Kiss, who and why is the woman in Tomoe's arms right before he goes on search of Dragon King's eye?

This happens when Nanami first goes in past and sees Tomoe caring for the "beautiful woman".

Is the woman Yukiji? If yes, why does he care for Yukiji as if she is Nanami? Can't he differentiate between the two?

I have only watched the anime and not the manga. Spoilers are fine for me.
As you've said, this happened in the past, 500 or so years before Nanami was even born.  So why should it be odd for him to care for another woman besides Nanami?  It doesn't seem an odd situation to me. 

But odd you should ask this question, because the spoiler answer is: Yes, it is Yukiji whom he was carrying for in the past, and actually, no, he can't entirely tell her and Nanami apart. As implied by the fact that the Dragon King's eye was in Nanami's body in the present, Nanami is Yukiji's descendent, and looks quite a bit like her. 

But furthermore, further along in the manga, Nanami goes again into Tomoe's past in order to learn about him (for reasons I won't go into here.) After having seen him with Yukiji in the past, she knows that he believed himself in love Yukiji, and so she tries to not interfere with his relationship with Yukiji. It turns out that Yukiji hated youkai, and Nanami ends up saving Tomoe's life when Yukiji refuses. In order to not interfere, Nanami tries to hide herself and tells people to tell Tomoe it was Yukiji that saved him. So, he had in fact fallen in love with Nanami, but due to misdirection, thought it was Yukiji. It's implied on some level he's aware that they are actually different women, as he seems a little confused when they act differently.