From the in-game books I've read I've gotten the impression that most Nords joined the war in 174, so I think it's likely that Ulfric did, too. My guess is that he's around 45, give or take a year or two.
The Great War states that "fresh legions from Skyrim" arrived in the Imperial City in 173; when Titus abandoned the city in 174 and marched north he met up with even more reinforcements marching south from Skyrim. However that doesn't mean that Nords weren't already joining and serving in the Legions from the outset of the war, in fact given the prevalence of Nords in the Imperial military I'm betting quite a few of them were already serving when the war broke out and that even more started enlisting soon after the Dominion attacked.
I think it's unlikely that Ulfric didn't join until 174 because the Imperial City was sacked in 174 and we know for a fact that he was captured by the Thalmor
before that ever happened. If he didn't join until that late and was subsequently captured that quickly, it doesn't give him much time to rack up all the combat experience he supposedly has and make him the seasoned Legion veteran that he supposedly is (I've seen references in various forums to him having been a "commander" in the Legion during the GW, if someone can point me to an in-game source that confirms this, much obliged). This is a guy who was leading a Nord militia against the Forsworn in the Reach and defeating them handily (the Thu'um helped, but still) within two years of the end of his Legion service, so I think it's unlikely that he spent ten years navel-gazing at HH, joined the Legion and fought for a few months and was captured, and then suddenly emerges as such a force to be reckoned with as a military leader.
And also his experiences in the war itself, unrelated to what happened in Thalmor custody, seem to have left some real scars on him. I'm not saying that can't happen in combat in a few weeks or even a few days if things are ugly enough, but the way he talks about the dying men he held in his arms, that sounds like somebody who was in the **** for a while and in a position of some responsibility as well. I don't see that happening if he joined that late in the game and only served for a few months before becoming a POW.