Arbit must be recited on Mosa’eh Yom Kippur as it is every other night of the year; the recitation of the Ne’ila prayer at the end of Yom Kippur does not absolve one of the obligation to pray Arbit.
The Arbit prayer must be recited slowly, and not in a rush as though one is rushing to end the fast like a child running home at the end of the school day. Rushing through the Arbit prayer after Yom Kippur gives the Satan the opportunity to argue against us before G-d, and show how we hurry to finish Yom Kippur. On the other hand, the Hazan must ensure not to prolong the prayer service, out of consideration for the fact that the congregants are weary from the fast. The prayer should be conducted at a reasonable, respectable pace, neither slowly nor quickly.
Our custom is to begin Arbit as we do on ordinary weeknights, with the three verses of "Hashem Seva-ot," followed by half-Kaddish and "Ve’hu Rahum." We add "Ata Honantanu" in the "Ata Honen" section of the Amida just as we do on Mosa’eh Shabbat.
If one mistakenly recited during Arbit "Ha’melech Ha’kadosh" or "Ha’malech Ha’mishpat," as we do during the Ten Days of Repentance, rather than changing back to "Ha’Kel Ha’kadosh" and "Melech Ohev Sedaka U’mishpat," he does not have to repeat the Amida. This is the view of the majority of the Halachic authorities. Nevertheless, in order to satisfy all opinions, Hacham Ovadia Yosef writes that one should preferably repeat the Amida and stipulate that according to the view that the repetition is not necessary, the prayer should be considered a voluntary prayer.
Just as one must add some time onto Yom Kippur at the beginning of the holiday, one must add some time at the end, as well. There is no set amount that must be added, and it suffices to add just a few minutes after nightfall onto Yom Kippur.