You might know that every time numbers appear in Japanese, they are usually accompanied by a counter word, e.g. つ, 個, 回, 番, ...
To say "three items" you can say 3つ, "3 pieces" is 3個, "three times" is 3回, "number three" is 3番.
To make the number into an ordinal, e.g. "three" into "third", you simply add 目 as so
3つ目 third item
3個目 third piece
3番目 third [number]
3回目 third time
So, you example first example sentence become
これで3回目です。
This is the third time.
Alas, being n th place (e.g., in a tournament) already has its own counter. In Japanese, you don't need an ordinal, but just the counter 位{い}, so your second example becomes
私は世界3位の選手です。
I am the third best player in the world.