Po čtyřech letech zase začal pít.
"He started to drink again after four years." is identical in meaning to "After four years, he started drinking again."
"After four years, he started to drink again." is not possible?
It is possible and accepted.
Unfortunately we have no report from you so we have no idea what might be the issue, if you made a typo or if it is a Duolingo bug...
Why not: "after four years he began to drink again"?
"After four years, he began to drink again" is also accepted.
Is there a solid rule about the usage of "let" vs "rok"? I thought it was the rule of 5 or more, i.e., čtyři roky, pět let. But that doesn't seem to be the case.
No, there isn't, it wil depend on the case, context... Better observe the actual usage. "pět roků" is completely correct as well.
what is wrong with " after four years he again started drinking"
It sounds quite unnatural in to me (native AmE). Adverbs generally go after the verb.