It it okay/enough to disinfect newly bought bucket & inner pump with a dish detergent like Fairy?

As long as it is thoroughly washed out you are fine. I used HCl (1M concentration) to remove some buildup on buckets before. Made quick work of what probably would have teken 30-60mins of hard scrubbing. Gotta step away from the fumes, though, bwahaha. Obviously, not an option for a pump. I'd refer to the pump manual on maintenance. Make sure you don't use some chemical that destroys the interior parts. cracking it open and taking a soapy brush to everything is probably enough.

You need hydrogen perroxide