This is pretty spoiler heavy if you haven't sided with Ulfric in the civil war.

When I enter Solitude, a Nord is about to get beheaded. Role-playing my current character, I would presume the guy's innocence and want to intervene. However, I cannot take down the whole city myself, not yet anyway. Assuming that the Stormcloaks eventually attack Solitude, will you be able to save that guy because you will have Stormcloak help?

Nope, he will still die even if Stormcloaks take the city and you never enter Solitude before.

So if I want him to live, I have to come in guns blazing before Ulfric attacks Solitude?

He'll die anyway if you do manage to save him. It's scripted.

This. Someone posted a link to a youtube video last week (maybe 2 weeks ago? I forget) where somebody was doing a tutorial on how to save the guy. After the guards were dead and the townspeople calmed down, the guy walked around the chopping block and just fell over dead.