I'm under H1B as well.
I got my American Express card in the first week in US, even before getting
my SSN. And I got the Citibank Thankyou Premier card in a month.
For Amex, I used global card transfer service, and they let me choose any
cards that are available, except Centurion. The process was pretty simple.
Everything was done over the phone, and I didn't have to submit any
documents.
As for Citi, you just need to talk to your banker. There is a separate
channel for global customers to apply credit cards. The welcome gifts are a
bit better than those in citi.com. You have only 4 cards (Citi Executive/TY
Premier/TY Preferred/AA Platinum Select) to choose from though. Citi
requires a number of documents which is much troublesome than Amex.
Both gave me a high credit limit, roughly equivalent to 1.5 to 2 times
monthly salary.