If I buy this 1 month card from Saturn (see picture) can I start playing online or do I have to provide account details (PayPal / credit card)?
Yes and show your ID.
My mother is about to go to town, she could buy it for me.
Since it's a subscription, you (as far as I know) still have to enter a payment method
No problem.
Oh yes, you need a passport or your ID which you then have to enter in the Playstation Store (at least that's how it is with me) PS. You have to be 18+ Just take it from your mother if you are still a minor.
Otherwise have fun playing.