Alfie is my first cat and I too was very anxious about his behaviour when I first got him.
He wouldn't come out of his carry-case for the first 6 hours or so (though he did accept a treat).
When he finally came out he accepted some food and water and seemed happy enough to be stroked, but at the slightest noise he disappeared behind the curtain and had to be gently coaxed out.
My major worry was that he did not go to the toilet for over 24 hours. I was beside myself with worry and ended up phoning an emergency line for a local vet. On the Vet's advice I brought him in, knowing that it was going to cost £100 just for them to see him (this was a Sunday). To my horror she basically told me that she reckoned he probably suffered from blocked bladder syndrome and whoever had handed him in to the rescue centre had "pulled a fast one - and she'd seen it so many times before". She asked it I was "terribly attached to him?" and suggested that I take him back to the rescue centre in the morning. Also suggested that I would be better off with a female cat!
I was devastated! I'd only had him a day and yet I was hurting like the vet had told me my cat of 20 years had to be put to sleep.
Anyway, on returning to my house, he hopped into the litterbox and 'let it all out' so to speak and hasn't looked back since!
That scared little scrawny cat is now the most chilled, relaxed moggy you could meet. You could set off a bomb next to him and he'd barely blink!