I have several question regarding my tax status and FICA responsibility. Here's my case
-I was a student on F-1 visa until May 2006.
-Started working on OPT (still F-1 visa) in May 2006. I was not paying FICA taxes during this time.
- Received my H-1B visa in October but my company did not start withholding the FICA tax.

Since I was in the US for 31 days after my exemption for substantial presence test was over by virtue of the visa status change, I assume I would be considered a resident for tax purpose.

Q1. Am I responsible for FICA tax only after I received the visa or for the whole year?
Q2. How do I pay the tax when my employer did not withhold it, considering they are responsible for a part of it?

You are a resident from the date your H-1B started, but you must file either a non-resident alien tax return (Form 1040NR or 1040NR-EZ) or a dual-status tax return (BOTH Form 1040 and 1040NR).

1) You are liable for Social Security and Medicare the day your H-1B visa became effective.

2) You can do nothing. The EMPLOYER must pay the FICA taxes and must withhold your share from your salary. If they fail to do so, the IRS will hold the EMPLOYER accountable, not you. There is no mechanism for you to pay your share of these taxes direct to the IRS; it must be paid through the employer.