As far as the cost, see the "What does it cost?" question in the FAQ section of:
Apple Arcade - Apple
I doubt if you can get a free trial without providing a payment method – since, if you are eligible for the free trial and take it, "Plan automatically renews after trial until cancelled" (small print at bottom of same page).
If you follow the chain of links, you eventually get to a Terms and Conditions document that says, in part,
"Upon claiming the offer, you agree to a $6.99/month subscription that starts immediately after the free offer period and automatically renews until cancelled. You can cancel at any time in Settings at least one day before each renewal date. If you cancel during your free offer period, you and your family members will immediately lose access to Apple Arcade and the remainder of your free trial. You can’t reactivate this trial."
https://www.apple.com/promo/pdf/EN_US_Arcade_Promo_TandCs.pdf
As for whether a credit card is required, or an Apple Account balance replenished with gift cards would be OK, I do not know. You'd have to ask Apple about that.