score:7
No, Milan Cathedral does not include a statue of Vladimir Putin.
I haven't been able to find any complete list of all statues on the building (there are, after all, literally thousands of them). But putting "putin statue milan cathedral", "milan duomo putin", and several other similar terms into my preferred search engine provided absolutely no results (not even from unrealiable-looking websites) to back up the idea that there's any statue or other representation of Putin.
The explanation that he's allegedly depicted as he made a "significant contribution" is implausible. Construction started in the 1300s and was completed in 1965. In 1965 Vladimir Putin turned 13 years old. Displays of religious belief were, shall we say, "heavily discouraged" in the Soviet Union. It's difficult to imagine how or why he would have made a significant contribution to a Roman Catholic cathedral in Milan (he is not a Catholic, like the vast majority of Christians in Russia he is a member of the Russian Orthodox church).
Of course a contribution could also be made later on to upkeep, not just construction. "Complete" is a loose term, there are apparently some marble blocks not yet carved into statues. But any contribution "significant" enough that they would carve a statue of the donor as part of the cathedral is surely a mind-bogglingly large sum. For a man with no obvious links to Milan and who belongs to an entirely different branch of Christianity to make such a contribution would be extremely difficult to explain. For him to somehow avoid any publicity from it despite his extremely public position, even more so.
All in all, the idea is absurd enough - and Putin a well-known enough public figure - that I consider the absence of evidence to be extremely compelling evidence of absence.