Meaning: To strike a bad patch means to go through a period of difficulty, failure, or misfortune.
The phrase is used in the plural here - "struck several bad patches" - so he faced repeated periods of difficulty.
The clause "before he made good" fixes the context as his career, because "to make good" means to become successful.
So the sentence means he faced many professional difficulties before he finally succeeded.
Why the other options are wrong:"Came across bad soil" is a literal misreading - "struck" here does not refer to land or soil.
"Went through many illness" is too narrow; the idiom refers to difficulty in general, not to illness.
"Had a bad time" is too vague and general - it ignores the career and success context created by "made good".
Hence "Had many professional difficulties" is the correct meaning.