So this is a question I have wanted to ask for a little while now: Do you think Ulfric is a hothead or a Cunning Politician?

I, for one, am going more for the Cunning than the Hothead and my reason for this is as follows:
I think it seems to be too much of a coinsidence that Ulfric started to speak against the Empire at the Moot where Torygg was elected and either sometime before and sometime after founded what would later become known as the Stormcloak Army.
By making sure that all people in the governing body of the land knew what he wanted and starting to prepare a military order he essentially lays the foundation for the rebellion. From here he, from what I understood, to continuing to speak against the Empire and uses the banning of Talos as a rallying point.

By doing these three, seemingly very simple, things he have spread his message to both the ruling class and the commoners and laid the foundation for an army the Jarls, other persons of power and the commoners could contribute to. And he crowns his work by slaying the High King in a duel that is legal under Nordic customs, with an ancient and revered Nordic form of magic; litterally splitting the land in half and forcing the Empire to take drastic action against him and his supporters(as much as the Empire could spare of soldiers and resources anyway, being busy guarding the border to the Aldmeri Dominion. Note: Tullius tells you this directly in the Season Unending quest when you try to persuade him to attend the peace council. A topic about why he haven't recieved more reinforcement).

Hot headed [censored] who is great at the emotional manipulation part of politics.

I don't think he's either.
I think he's earnest and believes he's doing the right thing and making the right choices, but I think of him as misguided. Some of the choices that he may sincerely believe to be good for Skyrim probably aren't. The only negative thing I think he's guilty of is being ambitious and having some arrogance in the sense of feeling that only he can fix things.

Given the choice between the two, I'd have to go with cunning politician.

If only because when you spend time with him in the game, he does nothing to indicate that he's any more of a "hothead" than any other NPC who might qualify for that moniker. He's no more inclined to angry outbursts than Balgruuf, for instance, who some people seem to revere without reservation.

And as you say, Tdroid, the current situation is not the result of some rash, impetuous decision to upset the apple cart. He's been Jarl for a while now, we're not sure how long, but it seems clear that he was already in that position when Torygg's father was High King. So he has been biding his time... either as a calculated measure to make sure all his ducks were in a row before he acted, or because he hoped that something would happen that would make such drastic action unnecessary or give him a reason to wait a little longer. A hothead wouldn't do that - he'd just wake up one day in a snit and ride off to Solitude and kill the High King and then go "uh oh" and try to cobble together a working rebellion on the fly because that had just become his only option.