1-Has he won a prize that I don't know of?
2-Has he won the lottery that I don't know of?
I think in the first one "that I don't know of" modifies "a prize" and not "his winning the prize". It does not really convey the meaning that it is supposed to convey.
The sentences are supposed to mean respectively:
a-Has he won a prize without me learning about it?
b-Has he won the lottery without me learning about it?
The second suggest he may have won a lottery and you are unaware of this lottery, which makes it a very odd question IMO.