Just started a game at immortal as Rome. I've been alternating depending on if I feel I want to challenge myself or not. Thought I was in good shape - friendly with England, and I had a perfect Petra city set for my 2nd city, with probably 15 desert hills in the area. England takes out a neighbouring city state. Okay, not a big deal, I can live with that. A few turns later, I notice a lot of archers and warriors nearby... Then I get the declaration of war.
Okay, I've fought off AI invasions before. They're also coming at multiple cities, so neither has enough to cause damage. And I actually had a handful of archers and warriors around as I was debating whether to go to legions to take her out. So I fight off the first wave of her troops with what I had, and switch my cities to build more warriors and archers just in case. Research quickly switches over to iron working to see how long before i can get my legions online.
And she keeps coming with troops. Like, quite a lot of them. But I just barely got a legion in time so that my cities defense rocket up, and I can hold her off. And then I have legions curb-stomping her cities when I bring along a battering ram. So I defended it, and ended up nearly wiping her out, and now I'm in a fairly solid space I think. Sadly, missed out on my chance to build Petra in that one city, so now it's stuck being a kind of crappy desert hill city. But I survived, albeit the only reason being that I was Rome and had a strong iron-free unit to use. I didn't have any Iron in my territory, so if I were any other civ I would have been easily beaten down.
So yeah, Immortal is tough. Would certainly recommend if you want to take the jump to take one of the better civs, and one of the ones with a strong early UU that you can use to defend and kickstart yourself. But yeah, when the AI comes, they come hard. Their tactics still need work, so you should still be able to defend in most instances, but be prepared to fight a long early battle.