Will they be able to walk-through it? If so, will they successfully cross over to the land of the dead, since they can't normally?
As I understand it, in the HP universe, Ghosts are not really the 'souls' of those they resemble -- they are more of an imprint of the consciousness that is left behind when some die. Similar to what is captured in Paintings and Photographs, actually.
Nick explained it in OotP:
“Wizards can leave an imprint of themselves upon the earth, to walk palely where their living selves once trod,” said Nick miserably. “But very few wizards choose that path.”
“I was afraid of death,” said Nick softly. “I chose to remain behind. I sometimes wonder whether I oughtn’t to have… well, that is neither here nor there… in fact, I am neither here nor there…” He gave a small sad chuckle. “I know nothing of the secrets of death, Harry, for I chose my feeble imitation of life instead. I believe learned wizards study the matter in the Department of Mysteries—”
Snape had a similar comment:
A ghost, as I trust that you are all aware by now, is the imprint of a departed soul left upon the earth... and of course, as Potter so wisely tells us, transparent.
I read this as an indication it's closest to an echo or a shadow of a soul, not an actual one; in each case, they use the term 'imprint.'
The Harry Potter wiki has this comment:
As ghosts are imprints of souls of the said deceased wizards and witches, they are unable to move beyond the veil, and are forever trapped in the world of living as intangible beings.
However, the canon status of it is questionable; it does seem consistent, though.
My guess (and that's all it is) would be that a ghost would either not be able to pass it, or would be destroyed; I have no canon backup, tho. Possibly you are right; their true 'self' is held to earth until the ghost passes on - if that is the case, then passing through the arch would allow them to pass on, although how much link the ghost has to the actual soul (assuming there is one) is also never stated.
Souls and the afterlife being of such core relevance to many religions, I think it's no accident that JKR left some of this ambiguous, the same way she mostly dodged religion showing up in the books. (Major Holidays are present.. but they have their secular aspects, and we never really see religious details.)
I hate giving a 'There's no answer, and short of WOG, we aren't supposed to be able to conclude one', but I think that's exactly the case here. I'll be delighted if someone can come with a JKR quote to refute that, tho. :)
Ghosts are transparent, moving, talking and thinking versions of wizards and witches who wished, for whatever reason, to remain on earth.
Tales of Beedle the Bard - page 70 - Bloomsbury - Babbity Rabbity and the Cackling Stump
KHW also mentions what Snape and Nearly Headless Nick say about the properties of ghosts.
Respectfully, though, I can find no canon source that addresses whether a ghost can go through the Veil or, if so, what would happen to them. We know more about the Love Room in the Department of Mysteries than we do about the Death Room. Aside from what we learn in Order of the Phoenix, I can't pinpoint a canon reference to the Death Room or the Veil, or a J.K. Rowling interview that discusses it or whether ghosts even can go through the Veil. As always please correct me if I am wrong.
So I think we've established what a ghost is, but even to make a thoughtful speculative answer is difficult because while we know about ghosts, we do not know what happens behind the Veil. Well, 1) at least some people who have seen death can hear muttering voices of the dead behind the Veil and 2) some of the newly dead can slip behind the Veil once their life force leaves them (like Sirius); I don't think a spirit of a deceased individual becomes a ghost until they reject "moving on".
If I were to make my own speculative observation, based on J.K. Rowling's encouragement to her fans to interpret the King's Cross chapter in a way that feels comfortable, I would imagine perhaps she might feel the same about the Veil -- that it's open to interpretation. But that's still not a canonical answer. I don't think this question is truly answerable without more information from J.K. Rowling about the Veil.