I believe the Rebellion was a low-level thing in Skyrim since the WGC. It wasn't until Ulfric challenged Torygg and won did the rebellion become a proper civil war.
I also think this was probably the case. IIRC there is dialogue from one NPC (can't remember who) who states that the secessionist movement was born in the aftermath of the Markarth Incident, which happened 25 years prior to the start of the game.
In this scenario, Tullius was already garrisoned in Skyrim, maybe specifically to get rid of the rebels or just as standard operating procedures for the Empire (who typically place garrisons in all provinces they control.) He failed to capture Ulfric at Solitude (since the Gate Guard let him out) but did manage to capture him in an ambush and had them sent to Helgen.
Can't agree with this as it's contradicted by in-game dialogue. Hadvar says that Tullius has "turned things around for the Empire" since his arrival a few months previously. I believe there is also other dialogue indicating that Tullius is a troubleshooter that the Emperor sends in to take over when things are going badly in a particular location or conflict. In other words, he wasn't in Skyrim when Torygg was killed and the current hostilities started, but things were going badly enough for the Legion in the initial stages of the war that Mede sent in one of his top guys to take over and turn it around.
It does indicate, though, that the war in its current "hot" state has been going on for several months at least, and presumably it's been that long since Torygg was killed. Ulfric wasn't captured on his way back to Windhelm from winning the duel in Solitude... which would seem unlikely anyway as Ralof says they were ambushed near Darkwater Crossing, and that's all the way down by Eastmarch's border with the Rift. More likely that Ulfric was traveling through Stormcloak held (and presumably secure) territory, either inspecting the troops or maybe paying a visit on his reluctant ally the Jarl of Riften - who knows?
At any rate, it seems pretty clear to me that Torygg's death was some months in the past when the game started, and that Tullius arrived some time after that and took over command of the Legion in Skyrim, then came by whatever information allowed him to plan and carry out the ambush that resulted in Ulfric's capture.