Known Plaintext Attack for Modified OFB Mode of Operation
It should be $256*128 = 4096$ bytes. The period at most 256 and each encryption outputs 128-bits.
When the $FB,FB,\ldots,FB$ series enter the plaintext, again we are taking the 8 MSB form the ciphertext that has still 256 different possibilities. To see the picture clearly;


In the first case the input space has only 256 plaintexts consist of 120-zero and the 8-bit.
In the second case the input space still has 256 plaintexts consist of repeating bytes. $$00\ldots 00, 01\ldots 01, \ldots , FF\ldots FF$$ Again, once $$FB_i = FB_j, i \neq j$$ we will have a cycle with at most 256 period.