It's true that he might not consider a vampire to not be a "true nord" but he also probably could not deny that there would be benefits to being an immortal High-King who can literally feast on the blood of his Imperial foes.

Not to mention Ulfric's appearance kinda screams vampire more than it does Viking anyway.

Prime vampire canidate and you are right he does look like a vampire.

Probably not. If people found out, it would be a [censored]-storm. The Temple Faithful would flee or take up arms, the Vigilant would have an actual cause, and even his closest supporters would turn on him if they're not down with the vamp stuff.
It would be eggshells, and Ulfric isn't as clever as The Lord from Oblivion to pull it off unnoticed.

No. He would not end up in his beloved sovngarde if he became a vampire. And if he was a vampire even more people would hate him.