I'm trying to learn what is grammatically correct given the parameters I've provided. It depends on what you are counting. If you are counting groups and there is only one group, then you need "New arrival this week". If there is more than one group, then you need "New arrivals this week." It's as simple as that.

1. In terms of online shopping, 'new arrivals' and 'new products' mean exactly the same thing and you can use them interchangeably. That is, they mean new things that the shop didn't previously sell. It doesn't mean something that they have additional stock for something that they already sell.