Is it still possible to get the NVIDIA GeForce 8600M GT repaired through the repair extension program for my Macbook Pro?

That program ended on Dec 7,2012. You might try to plead your case, nicely, to your local Apple store or Authorized Service Provider, to see if they will, on a complementary basis, do the repair.

I just ran into the same situation when my GPU went out over Christmas. I was unable to get the free replacement since the program is over, but ask about Apple's flat rate repair pricing.

It cost me $310 + tax, but they took the laptop, sent it out to replace the logic board, cleaned it up REAL nice, and shipped directly back to my home. Just the amount I was willing to put back into this 4 year old laptop; I'm content :-)