You should not use the in
The war campaign has shot up Putin's ratings.
Yes, the noun "ratings" is definite, but it already has a word that indicates whose ratings they are: Putin's ratings.
You may think of it as
The war campaign has shot up his ratings.
Words like "his, her, their" are called "possessive determiners".
The definite article "the" is also a determiner. Some determiners can be combined, but we don't combine "the" with possessive determiners:
I love to eat the her cakes. (WRONG!)
If you want to know which determiners can be combined, see Wikipedia's article English determiners, subsection Combination of determiners.
(Note that I did not write "see the Wikipedia's article 'English determiners'")