Does the Hamlet Fortinbras describes (5.2.339-344) sound like the Hamlet we have known? What will happen to the kingdom under...

In my edition of Shakespeare's
Hamlet, the line numbers you give feature Hamlet speaking before he
dies.  In my edition, I think the lines you ask about are 380-388, in which Fortinbras
says:

...Let
four captains

Bear Hamlet like a soldier to the
stage,

For he was likely, had he been put
on,

To have proved most
royal,...

Essentially, all
Fortinbras says is that if Hamlet had been crowned king, he likely would have proved
"most royal." 

What being royal consists of is probably a
matter of opinion, but in general, Hamlet, throughout the play, is noble, is fully aware
of the fact that actions have consequences, shows leadership abilities, apparently is an
excellent swordsman, and is a good man, at least for the most part and when it comes to
the larger issues of a kingdom.  These traits suggest Fortinbras is correct, and Hamlet
would have been most royal.

Concerning what happens in
Denmark with Fortinbras in control, any answer I give would be purely specuation and
irrelevant to the play.  Finally, Fortinbras doesn't really have any revenge to get,
since King Hamlet, the leader of the forces in the battle during which King Norway was
killed, died long ago.