You can say either, and the meaning is exactly the same.
1. I heard somewhere that 'pity' is outdated or only British word. Is it right?
No, that's not true. 'It's a pity' is not at all outdated - we use this expression all the time in modern English. And it's not 'only' a British word. It is also used elsewhere.
2. If I say 'it's a shame', wouldn't such sentence have the shade of main definition of the word 'shame' and hurt feelings of the person I talk to?
No, not at all. For English speakers today, there is no connection in our minds between the sense of shame, and the set phrase 'It's a shame'.